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MCQ (Single Correct Answer)

JEE Main 2024 (Online) 1st February Evening Shift
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of the tenth and the fifth terms is $15: 7...
JEE Main 2024 (Online) 1st February Morning Shift
Let $3, a, b, c$ be in A.P. and $3, a-1, b+1, c+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :
JEE Main 2024 (Online) 31st January Evening Shift
Let $$2^{\text {nd }}, 8^{\text {th }}$$ and $$44^{\text {th }}$$ terms of a non-constant A. P. be respectively the $$1^{\text {st }}, 2^{\text {nd }}...
JEE Main 2024 (Online) 31st January Morning Shift
For $$0 (I) If $$\alpha \in(-1,0)$$, then $$b$$ cannot be the geometric mean of $a$ and $$c$$ (II) If $$\alpha \in(0,1)$$, then $$b$$ may be the geom...
JEE Main 2024 (Online) 31st January Morning Shift
The sum of the series $$\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots$$ up to 10 -terms is
JEE Main 2024 (Online) 30th January Evening Shift
Let $$a$$ and $$b$$ be be two distinct positive real numbers. Let $$11^{\text {th }}$$ term of a GP, whose first term is $$a$$ and third term is $$b$$...
JEE Main 2024 (Online) 30th January Morning Shift
Let $$S_n$$ denote the sum of first $$n$$ terms of an arithmetic progression. If $$S_{20}=790$$ and $$S_{10}=145$$, then $$\mathrm{S}_{15}-\mathrm{S}_...
JEE Main 2024 (Online) 29th January Evening Shift
If $$\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}$$ are in an A.P. and $$\log _e \mathrm{a}-\log _e 2 \mathrm{~b}, \log _e 2 \mathrm{~b...
JEE Main 2024 (Online) 29th January Evening Shift
If each term of a geometric progression $$a_1, a_2, a_3, \ldots$$ with $$a_1=\frac{1}{8}$$ and $$a_2 \neq a_1$$, is the arithmetic mean of the next tw...
JEE Main 2024 (Online) 29th January Morning Shift
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to...
JEE Main 2024 (Online) 29th January Morning Shift
In an A.P., the sixth term $$a_6=2$$. If the product $$a_1 a_4 a_5$$ is the greatest, then the common difference of the A.P. is equal to
JEE Main 2024 (Online) 27th January Evening Shift
$$\text { The } 20^{\text {th }} \text { term from the end of the progression } 20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1...
JEE Main 2024 (Online) 27th January Morning Shift
The number of common terms in the progressions $4,9,14,19, \ldots \ldots$, up to $25^{\text {th }}$ term and $3,6,9,12, \ldots \ldots$, up to $37^{\te...
JEE Main 2023 (Online) 15th April Morning Shift
Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}, G_{3}$ be three geometric means of two distinct positive numbers. Then $G_{1}^{4}+G...
JEE Main 2023 (Online) 13th April Evening Shift
Let a$$_1$$, a$$_2$$, a$$_3$$, .... be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2 and the product of its 3rd and...
JEE Main 2023 (Online) 13th April Morning Shift
Let $$s_{1}, s_{2}, s_{3}, \ldots, s_{10}$$ respectively be the sum to 12 terms of 10 A.P. s whose first terms are $$1,2,3, \ldots .10$$ and the commo...
JEE Main 2023 (Online) 12th April Morning Shift
Let $$ $$ be a sequence such that $$a_{1}+a_{2}+\ldots+a_{n}=\frac{n^{2}+3 n}{(n+1)(n+2)}$$. If $$28 \sum_\limits{k=1}^{10} \frac{1}{a_{k}}=p_{1} p_{2...
JEE Main 2023 (Online) 11th April Evening Shift
Let $$a, b, c$$ and $$d$$ be positive real numbers such that $$a+b+c+d=11$$. If the maximum value of $$a^{5} b^{3} c^{2} d$$ is $$3750 \beta$$, then t...
JEE Main 2023 (Online) 11th April Morning Shift
Let $$x_{1}, x_{2}, \ldots, x_{100}$$ be in an arithmetic progression, with $$x_{1}=2$$ and their mean equal to 200 . If $$y_{i}=i\left(x_{i}-i\right)...
JEE Main 2023 (Online) 10th April Evening Shift
If $$\mathrm{S}_{n}=4+11+21+34+50+\ldots$$ to $$n$$ terms, then $$\frac{1}{60}\left(\mathrm{~S}_{29}-\mathrm{S}_{9}\right)$$ is equal to :
JEE Main 2023 (Online) 10th April Morning Shift
Let the first term $$\alpha$$ and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms i...
JEE Main 2023 (Online) 8th April Evening Shift
Let $$\mathrm{a}_{\mathrm{n}}$$ be the $$\mathrm{n}^{\text {th }}$$ term of the series $$5+8+14+23+35+50+\ldots$$ and $$\mathrm{S}_{\mathrm{n}}=\sum_\...
JEE Main 2023 (Online) 8th April Morning Shift
Let $$S_{K}=\frac{1+2+\ldots+K}{K}$$ and $$\sum_\limits{j=1}^{n} S_{j}^{2}=\frac{n}{A}\left(B n^{2}+C n+D\right)$$, where $$A, B, C, D \in \mathbb{N}$...
JEE Main 2023 (Online) 6th April Evening Shift
If $$\operatorname{gcd}~(\mathrm{m}, \mathrm{n})=1$$ and $$1^{2}-2^{2}+3^{2}-4^{2}+\ldots . .+(2021)^{2}-(2022)^{2}+(2023)^{2}=1012 ~m^{2} n$$ then $...
JEE Main 2023 (Online) 6th April Morning Shift
The sum of the first $$20$$ terms of the series $$5+11+19+29+41+\ldots$$ is :
JEE Main 2023 (Online) 1st February Evening Shift
The sum $$\sum\limits_{n = 1}^\infty {{{2{n^2} + 3n + 4} \over {(2n)!}}} $$ is equal to :
JEE Main 2023 (Online) 1st February Morning Shift
The sum of 10 terms of the series $${1 \over {1 + {1^2} + {1^4}}} + {2 \over {1 + {2^2} + {2^4}}} + {3 \over {1 + {3^2} + {3^4}}}\, + \,....$$ is...
JEE Main 2023 (Online) 31st January Evening Shift
Let $a_1, a_2, a_3, \ldots$ be an A.P. If $a_7=3$, the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero, then $n !-4 a_{n(n+2)}...
JEE Main 2023 (Online) 31st January Morning Shift
If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is...
JEE Main 2023 (Online) 30th January Evening Shift
Let $a, b, c>1, a^3, b^3$ and $c^3$ be in A.P., and $\log _a b, \log _c a$ and $\log _b c$ be in G.P. If the sum of first 20 terms of an A.P., whose f...
JEE Main 2023 (Online) 30th January Morning Shift
If $${a_n} = {{ - 2} \over {4{n^2} - 16n + 15}}$$, then $${a_1} + {a_2}\, + \,....\, + \,{a_{25}}$$ is equal to :
JEE Main 2023 (Online) 24th January Morning Shift
For three positive integers p, q, r, $${x^{p{q^2}}} = {y^{qr}} = {z^{{p^2}r}}$$ and r = pq + 1 such that 3, 3 log$$_yx$$, 3 log$$_zy$$, 7 log$$_xz$$ a...
JEE Main 2022 (Online) 29th July Evening Shift
$$ \begin{aligned} &\text { Let }\left\{a_{n}\right\}_{n=0}^{\infty} \text { be a sequence such that } a_{0}=a_{1}=0 \text { and } \\\\ &a_{n+2}=3 a_{...
JEE Main 2022 (Online) 28th July Morning Shift
Consider the sequence $$a_{1}, a_{2}, a_{3}, \ldots$$ such that $$a_{1}=1, a_{2}=2$$ and $$a_{n+2}=\frac{2}{a_{n+1}}+a_{n}$$ for $$\mathrm{n}=1,2,3, \...
JEE Main 2022 (Online) 27th July Evening Shift
Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be $$\frac{98}{25}$$. The...
JEE Main 2022 (Online) 27th July Morning Shift
Suppose $$a_{1}, a_{2}, \ldots, a_{n}$$, .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of...
JEE Main 2022 (Online) 26th July Morning Shift
Consider two G.Ps. 2, 22, 23, ..... and 4, 42, 43, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is $${(2)^{{{225...
JEE Main 2022 (Online) 25th July Evening Shift
The sum $$\sum\limits_{n = 1}^{21} {{3 \over {(4n - 1)(4n + 3)}}} $$ is equal to
JEE Main 2022 (Online) 30th June Morning Shift
The value of $$1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}}$$ is equal to:
JEE Main 2022 (Online) 29th June Evening Shift
The sum of the infinite series $$1 + {5 \over 6} + {{12} \over {{6^2}}} + {{22} \over {{6^3}}} + {{35} \over {{6^4}}} + {{51} \over {{6^5}}} + {{70} \...
JEE Main 2022 (Online) 29th June Morning Shift
Let $$\{ {a_n}\} _{n = 0}^\infty $$ be a sequence such that $${a_0} = {a_1} = 0$$ and $${a_{n + 2}} = 2{a_{n + 1}} - {a_n} + 1$$ for all n $$\ge$$ 0. ...
JEE Main 2022 (Online) 28th June Evening Shift
If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of...
JEE Main 2022 (Online) 28th June Morning Shift
Let A1, A2, A3, ....... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = $${1 \over {1296}}$$ and A2 + A4 = $${7 \over {...
JEE Main 2022 (Online) 27th June Evening Shift
Let $$S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} + \,.....$$. Then 4S is equal to
JEE Main 2022 (Online) 27th June Evening Shift
If a1, a2, a3 ...... and b1, b2, b3 ....... are A.P., and a1 = 2, a10 = 3, a1b1 = 1 = a10b10, then a4 b4 is equal to -...
JEE Main 2022 (Online) 27th June Morning Shift
$$x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } } $$, where a, b, c are in A...
JEE Main 2022 (Online) 26th June Evening Shift
If $$A = \sum\limits_{n = 1}^\infty {{1 \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}} $$ and $$B = \sum\limits_{n = 1}^\infty {{{{{( - 1)}^n}} \...
JEE Main 2022 (Online) 25th June Evening Shift
The sum 1 + 2 . 3 + 3 . 32 + ......... + 10 . 39 is equal to :
JEE Main 2022 (Online) 24th June Evening Shift
Let x, y > 0. If x3y2 = 215, then the least value of 3x + 2y is
JEE Main 2022 (Online) 24th June Morning Shift
If $$\{ {a_i}\} _{i = 1}^n$$, where n is an even integer, is an arithmetic progression with common difference 1, and $$\sum\limits_{i = 1}^n {{a_i} = ...
JEE Main 2021 (Online) 1st September Evening Shift
Let Sn = 1 . (n $$-$$ 1) + 2 . (n $$-$$ 2) + 3 . (n $$-$$ 3) + ..... + (n $$-$$ 1) . 1, n $$\ge$$ 4.The sum $$\sum\limits_{n = 4}^\infty {\left( {{{2...
JEE Main 2021 (Online) 1st September Evening Shift
Let a1, a2, ..........., a21 be an AP such that $$\sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}} $$. If the sum of this AP is 1...
JEE Main 2021 (Online) 31st August Evening Shift
Let a1, a2, a3, ..... be an A.P. If $${{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} \over {{p^2}}}$$, p $$\ne$$ 10...
JEE Main 2021 (Online) 31st August Morning Shift
The sum of 10 terms of the series $${3 \over {{1^2} \times {2^2}}} + {5 \over {{2^2} \times {3^2}}} + {7 \over {{3^2} \times {4^2}}} + ....$$ is :...
JEE Main 2021 (Online) 31st August Morning Shift
Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmet...
JEE Main 2021 (Online) 27th August Evening Shift
If 0 < x < 1 and $$y = {1 \over 2}{x^2} + {2 \over 3}{x^3} + {3 \over 4}{x^4} + ....$$, then the value of e1 + y at $$x = {1 \over 2}$$ is :...
JEE Main 2021 (Online) 27th August Morning Shift
If 0 < x < 1, then $${3 \over 2}{x^2} + {5 \over 3}{x^3} + {7 \over 4}{x^4} + .....$$, is equal to :
JEE Main 2021 (Online) 27th August Morning Shift
If for x, y $$\in$$ R, x > 0, y = log10x + log10x1/3 + log10x1/9 + ...... upto $$\infty$$ terms and $${{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ....
JEE Main 2021 (Online) 26th August Morning Shift
The sum of the series $${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$$...
JEE Main 2021 (Online) 26th August Morning Shift
If the sum of an infinite GP a, ar, ar2, ar3, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, ..........
JEE Main 2021 (Online) 25th July Morning Shift
Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of $${{{S_{4n}}} \over {{S_{2n}}}}$$ is :...
JEE Main 2021 (Online) 22th July Evening Shift
Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 $$-$$ S6 is equal to:...
JEE Main 2021 (Online) 20th July Evening Shift
If sum of the first 21 terms of the series $${\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + .......$$, where x > 0 is 504, th...
JEE Main 2021 (Online) 18th March Evening Shift
Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 $$-$$...
JEE Main 2021 (Online) 18th March Morning Shift
If $$\alpha$$, $$\beta$$ are natural numbers such that 100$$\alpha$$ $$-$$ 199$$\beta$$ = (100)(100) + (99)(101) + (98)(102) + ...... + (1)(199), then...
JEE Main 2021 (Online) 18th March Morning Shift
$${1 \over {{3^2} - 1}} + {1 \over {{5^2} - 1}} + {1 \over {{7^2} - 1}} + .... + {1 \over {{{(201)}^2} - 1}}$$ is equal to
JEE Main 2021 (Online) 26th February Evening Shift
The sum of the series $$\sum\limits_{n = 1}^\infty {{{{n^2} + 6n + 10} \over {(2n + 1)!}}} $$ is equal to :
JEE Main 2021 (Online) 26th February Morning Shift
The sum of the infinite series $$1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{22} \over {{3^5}}} + ......$$ i...
JEE Main 2021 (Online) 26th February Morning Shift
In an increasing geometric series, the sum of the second and the sixth term is $${{25} \over 2}$$ and the product of the third and fifth term is 25. T...
JEE Main 2021 (Online) 25th February Evening Shift
The minimum value of $$f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$$, where a, $$x \in R$$ and a > 0, is equal to :
JEE Main 2021 (Online) 25th February Morning Shift
If $$0 < \theta ,\phi < {\pi \over 2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}...
JEE Main 2020 (Online) 6th September Evening Slot
The common difference of the A.P. b1, b2, … , bm is 2 more than the common difference of A.P. a1, a2, …, an. If a40 = –159, a100 = –399 and b100 = a...
JEE Main 2020 (Online) 6th September Morning Slot
Let a , b, c , d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then :...
JEE Main 2020 (Online) 5th September Evening Slot
If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum ...
JEE Main 2020 (Online) 5th September Evening Slot
If the sum of the first 20 terms of the series $${\log _{\left( {{7^{1/2}}} \right)}}x + {\log _{\left( {{7^{1/3}}} \right)}}x + {\log _{\left( {{7^{1...
JEE Main 2020 (Online) 5th September Morning Slot
If 210 + 29.31 + 28 .32 +.....+ 2.39 + 310 = S - 211, then S is equal to :...
JEE Main 2020 (Online) 5th September Morning Slot
If $${3^{2\sin 2\alpha - 1}}$$, 14 and $${3^{4 - 2\sin 2\alpha }}$$ are the first three terms of an A.P. for some $$\alpha $$, then the sixth terms o...
JEE Main 2020 (Online) 4th September Evening Slot
The minimum value of 2sinx + 2cosx is :
JEE Main 2020 (Online) 4th September Evening Slot
Let a1, a2, ..., an be a given A.P. whose common difference is an integer and Sn = a1 + a2 + .... + an. If a1 = 1, an = 300 and 15 $$ \le $$ n $$ \le ...
JEE Main 2020 (Online) 4th September Morning Slot
If 1+(1–22.1)+(1–42.3)+(1-62.5)+......+(1-202.19)= $$\alpha $$ - 220$$\beta $$, then an ordered pair $$\left( {\alpha ,\beta } \right)$$ is equal to:...
JEE Main 2020 (Online) 3rd September Evening Slot
If the sum of the series 20 + 19$${3 \over 5}$$ + 19$${1 \over 5}$$ + 18$${4 \over 5}$$ + ... upto nth term is 488 and the nth term is negative, then ...
JEE Main 2020 (Online) 3rd September Morning Slot
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. ...
JEE Main 2020 (Online) 2nd September Evening Slot
If the sum of first 11 terms of an A.P., a1, a2, a3, .... is 0 (a $$ \ne $$ 0), then the sum of the A.P., a1 , a3 , a5 ,....., a23 is ka1 , where k is...
JEE Main 2020 (Online) 2nd September Evening Slot
Let S be the sum of the first 9 terms of the series : {x + k$$a$$} + {x2 + (k + 2)$$a$$} + {x3 + (k + 4)$$a$$} + {x4 + (k + 6)$$a$$} + .... where a $...
JEE Main 2020 (Online) 2nd September Morning Slot
If |x| < 1, |y| < 1 and x $$ \ne $$ y, then the sum to infinity of the following series (x + y) + (x2+xy+y2) + (x3+x2y + xy2+y3) + .......
JEE Main 2020 (Online) 2nd September Morning Slot
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :
JEE Main 2020 (Online) 9th January Evening Slot
Let an be the nth term of a G.P. of positive terms. $$\sum\limits_{n = 1}^{100} {{a_{2n + 1}} = 200} $$ and $$\sum\limits_{n = 1}^{100} {{a_{2n}} = 10...
JEE Main 2020 (Online) 9th January Morning Slot
The product $${2^{{1 \over 4}}}{.4^{{1 \over {16}}}}{.8^{{1 \over {48}}}}{.16^{{1 \over {128}}}}$$ ... to $$\infty $$ is equal to :
JEE Main 2020 (Online) 8th January Evening Slot
If the 10th term of an A.P. is $${1 \over {20}}$$ and its 20th term is $${1 \over {10}}$$, then the sum of its first 200 terms is...
JEE Main 2020 (Online) 8th January Morning Slot
Let ƒ : R $$ \to $$ R be such that for all x $$ \in $$ R (21+x + 21–x), ƒ(x) and (3x + 3–x) are in A.P., then the minimum value of ƒ(x) is...
JEE Main 2020 (Online) 7th January Evening Slot
Let $${a_1}$$ , $${a_2}$$ , $${a_3}$$ ,....... be a G.P. such that $${a_1}$$ < 0, $${a_1}$$ + $${a_2}$$ = 4 and $${a_3}$$ + $${a_4}$$ = 16. If...
JEE Main 2020 (Online) 7th January Evening Slot
If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to :
JEE Main 2020 (Online) 7th January Morning Slot
Five numbers are in A.P. whose sum is 25 and product is 2520. If one of these five numbers is -$${1 \over 2}$$ , then the greatest number amongst them...
JEE Main 2019 (Online) 12th April Evening Slot
If a1, a2, a3, ..... are in A.P. such that a1 + a7 + a16 = 40, then the sum of the first 15 terms of this A.P. is :...
JEE Main 2019 (Online) 12th April Morning Slot
For x $$\varepsilon $$ R, let [x] denote the greatest integer $$ \le $$ x, then the sum of the series $$\left[ { - {1 \over 3}} \right] + \left[ { - {...
JEE Main 2019 (Online) 12th April Morning Slot
Let Sn denote the sum of the first n terms of an A.P. If S4 = 16 and S6= – 48, then S10 is equal to :
JEE Main 2019 (Online) 10th April Evening Slot
The sum $$1 + {{{1^3} + {2^3}} \over {1 + 2}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 2 + 3}} + ...... + {{{1^3} + {2^3} + {3^3} + ... + {{15}^3}} \over...
JEE Main 2019 (Online) 10th April Evening Slot
Let $$a$$, b and c be in G.P. with common ratio r, where $$a$$ $$ \ne $$ 0 and 0 < r $$ \le $$ $${1 \over 2}$$ . If 3$$a$$, 7b and 15c are the firs...
JEE Main 2019 (Online) 10th April Evening Slot
Let a1, a2, a3,......be an A.P. with a6 = 2. Then the common difference of this A.P., which maximises the product a1a4a5, is : ...
JEE Main 2019 (Online) 10th April Morning Slot
The sum $${{3 \times {1^3}} \over {{1^3}}} + {{5 \times ({1^3} + {2^3})} \over {{1^2} + {2^2}}} + {{7 \times \left( {{1^3} + {2^3} + {3^3}} \right)} \...
JEE Main 2019 (Online) 10th April Morning Slot
If a1, a2, a3, ............... an are in A.P. and a1 + a4 + a7 + ........... + a16 = 114, then a1 + a6 + a11 + a16 is equal to : ...
JEE Main 2019 (Online) 9th April Evening Slot
Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls an...
JEE Main 2019 (Online) 9th April Evening Slot
If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its 11th term is :-
JEE Main 2019 (Online) 9th April Evening Slot
The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 +.... upto 11th term is :-
JEE Main 2019 (Online) 9th April Morning Slot
Let the sum of the first n terms of a non-constant A.P., a1, a2, a3, ..... be $$50n + {{n(n - 7)} \over 2}A$$, where A is a constant. If d is the comm...
JEE Main 2019 (Online) 8th April Evening Slot
The sum $$\sum\limits_{k = 1}^{20} {k{1 \over {{2^k}}}} $$ is equal to
JEE Main 2019 (Online) 8th April Evening Slot
If three distinct numbers a, b, c are in G.P. and the equations ax2 + 2bx + c = 0 and dx2 + 2ex + ƒ = 0 have a common root, then which one of the fo...
JEE Main 2019 (Online) 8th April Morning Slot
The sum of all natural numbers 'n' such that 100 < n < 200 and H.C.F. (91, n) > 1 is :
JEE Main 2019 (Online) 12th January Evening Slot
If the sum of the first 15 terms of the series $${\left( {{3 \over 4}} \right)^3} + {\left( {1{1 \over 2}} \right)^3} + {\left( {2{1 \over 4}} \right)...
JEE Main 2019 (Online) 12th January Evening Slot
If sin4$$\alpha $$ + 4 cos4$$\beta $$ + 2 = 4$$\sqrt 2 $$ sin $$\alpha $$ cos $$\beta $$; $$\alpha $$, $$\beta $$ $$ \in $$ [0, $$\pi $$], then cos($...
JEE Main 2019 (Online) 12th January Evening Slot
If   nC4, nC5 and nC6 are in A.P., then n can be :
JEE Main 2019 (Online) 12th January Morning Slot
Let  Sk = $${{1 + 2 + 3 + .... + k} \over k}.$$ If   $$S_1^2 + S_2^2 + .....\, + S_{10}^2 = {5 \over {12}}$$A,  then A i...
JEE Main 2019 (Online) 12th January Morning Slot
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an...
JEE Main 2019 (Online) 11th January Evening Slot
If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :
JEE Main 2019 (Online) 11th January Evening Slot
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)...
JEE Main 2019 (Online) 11th January Morning Slot
The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $${{27} \over {19}}$$.Then the common ratio ...
JEE Main 2019 (Online) 11th January Morning Slot
Let a1, a2, . . . . . ., a10 be a G.P.    If $${{{a_3}} \over {{a_1}}} = 25,$$ then $${{{a_9}} \over {{a_5}}}$$ equals...
JEE Main 2019 (Online) 10th January Evening Slot
Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k $$ \in $$ N (the set of natural numbe...
JEE Main 2019 (Online) 10th January Morning Slot
The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -
JEE Main 2019 (Online) 9th January Evening Slot
The sum of the following series $$1 + 6 + {{9\left( {{1^2} + {2^2} + {3^2}} \right)} \over 7} + {{12\left( {{1^2} + {2^2} + {3^2} + {4^2}} \right)} \o...
JEE Main 2019 (Online) 9th January Evening Slot
Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then $${a \ov...
JEE Main 2019 (Online) 9th January Morning Slot
If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be
JEE Main 2019 (Online) 9th January Morning Slot
Let $${a_1},{a_2},.......,{a_{30}}$$ be an A.P., $$S = \sum\limits_{i = 1}^{30} {{a_i}} $$ and $$T = \sum\limits_{i = 1}^{15} {{a_{\left( {2i - 1} \ri...
JEE Main 2018 (Online) 16th April Morning Slot
The sum of the first 20 terms of the series $$1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ...,$$ is :
JEE Main 2018 (Online) 16th April Morning Slot
Let $${1 \over {{x_1}}},{1 \over {{x_2}}},...,{1 \over {{x_n}}}\,\,$$ (xi $$ \ne $$ 0 for i = 1, 2, ..., n) be in A.P. such that x1=4 and x21 = 20. If...
JEE Main 2018 (Offline)
Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 12 + 2.22 + 32 + 2.42 + 52 + 2.62 ........... If B - 2A = ...
JEE Main 2018 (Offline)
Let $${a_1}$$, $${a_2}$$, $${a_3}$$, ......... ,$${a_{49}}$$ be in A.P. such that $$\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416$$ and $${a_9} + {a_...
JEE Main 2018 (Online) 15th April Evening Slot
If  a,   b,   c  are in A.P. and  a2,  b2,  c2 are in G.P. such that a < b < c and ...
JEE Main 2018 (Online) 15th April Evening Slot
Let    An = $$\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3}$$ $$-$$. . . . . + ($$-$$1)...
JEE Main 2018 (Online) 15th April Morning Slot
If x1, x2, . . ., xn and $${1 \over {{h_1}}}$$, $${1 \over {{h_2}}}$$, . . . , $${1 \over {{h_n}}}$$ are two A.P..s such that x3 = h2 = 8 and x8 = h7 ...
JEE Main 2018 (Online) 15th April Morning Slot
If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :
JEE Main 2017 (Online) 9th April Morning Slot
If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :
JEE Main 2017 (Online) 9th April Morning Slot
Let Sn = $${1 \over {{1^3}}}$$$$ + {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + ......... + {{1 + 2 + ....... + n}...
JEE Main 2017 (Online) 8th April Morning Slot
If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then $${{a + b} \over {a - b}}$$ is equal to :
JEE Main 2017 (Online) 8th April Morning Slot
If the sum of the first n terms of the series $$\,\sqrt 3 + \sqrt {75} + \sqrt {243} + \sqrt {507} + ......$$ is $$435\sqrt 3 ,$$ then n equals :...
JEE Main 2017 (Offline)
For any three positive real numbers a, b and c, 9(25$${a^2}$$ + b2) + 25(c2 - 3$$a$$c) = 15b(3$$a$$ + c). Then
JEE Main 2016 (Online) 10th April Morning Slot
Let z = 1 + ai be a complex number, a > 0, such that z3 is a real number. Then the sum 1 + z + z2 + . . . . .+ z11 is equal to : ...
JEE Main 2016 (Online) 10th April Morning Slot
Let a1, a2, a3, . . . . . . . , an, . . . . . be in A.P. If a3 + a7 + a11 + a15 = 72, then the sum of its first 17 terms is equal to :...
JEE Main 2016 (Online) 10th April Morning Slot
If   A > 0, B > 0   and    A + B = $${\pi \over 6}$$, then the minimum value of tanA + tanB is :
JEE Main 2016 (Online) 9th April Morning Slot
Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3is equal to : ...
JEE Main 2016 (Offline)
If the $${2^{nd}},{5^{th}}\,and\,{9^{th}}$$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :
JEE Main 2016 (Offline)
If the sum of the first ten terms of the series $${\left( {1{3 \over 5}} \right)^2} + {\left( {2{2 \over 5}} \right)^2} + {\left( {3{1 \over 5}} \righ...
JEE Main 2015 (Offline)
The sum of first 9 terms of the series. $${{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......$...
JEE Main 2015 (Offline)
If m is the A.M. of two distinct real numbers l and n $$(l,n > 1)$$ and $${G_1},{G_2}$$ and $${G_3}$$ are three geometric means between $$l$$ and n...
JEE Main 2014 (Offline)
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G....
JEE Main 2014 (Offline)
If $${(10)^9} + 2{(11)^1}\,({10^8}) + 3{(11)^2}\,{(10)^7} + ......... + 10{(11)^9} = k{(10)^9},$$, then k is equal to :
JEE Main 2013 (Offline)
The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is
AIEEE 2012
Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000. Statement-2: $$\sum\limits_...
AIEEE 2011
A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the savin...
AIEEE 2010
A person is to count 4500 currency notes. Let $${a_n}$$ denote the number of notes he counts in the $${n^{th}}$$ minute. If $${a_1}$$ = $${a_2}$$ = .....
AIEEE 2009
The sum to infinite term of the series $$1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....$$ is
AIEEE 2008
The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progressio...
AIEEE 2007
The sum of series $${1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......$$ upto infinity is
AIEEE 2007
In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is e...
AIEEE 2006
If $${{a_1},{a_2},....{a_n}}$$ are in H.P., then the expression $${{a_1}\,{a_2} + \,{a_2}\,{a_3}\, + .... + {a_{n - 1}}\,{a_n}}$$ is equal to
AIEEE 2006
Let $${a_1}$$, $${a_2}$$, $${a_3}$$.....be terms on A.P. If $${{{a_1} + {a_2} + .....{a_p}} \over {{a_1} + {a_2} + .....{a_q}}} = {{{p^2}} \over {{q^2...
AIEEE 2005
If $$x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,$$ where a...
AIEEE 2005
The sum of the series $$1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + .......$$ ad inf. is
AIEEE 2004
Let $${{T_r}}$$ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, $$m \ne n,\,\,{T_m} =...
AIEEE 2004
The sum of the first n terms of the series $${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1)}^2}} \over 2}$$ when n is ev...
AIEEE 2004
The sum of series $${1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........$$ is
AIEEE 2003
The sum of the serier $${1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..............$$ up to $$\infty $$ is equal to
AIEEE 2002
If 1, $${\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)$$ are in A.P. then x equals
AIEEE 2002
l, m, n are the $${p^{th}}$$, $${q^{th}}$$ and $${r^{th}}$$ term of a G.P all positive, $$then\,\left| {\matrix{ {\log \,l} & p & 1 \cr ...
AIEEE 2002
The value of $$\,{2^{1/4}}.\,\,{4^{1/8}}.\,{8^{1/16}}...\infty $$ is
AIEEE 2002
Fifth term of a GP is 2, then the product of its 9 terms is
AIEEE 2002
Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is
AIEEE 2002
$${1^3} - \,\,{2^3} + {3^3} - {4^3} + ... + {9^3} = $$

Numerical

JEE Main 2024 (Online) 1st February Evening Shift
If three successive terms of a G.P. with common ratio $\mathrm{r}(\mathrm{r}>1)$ are the lengths of the sides of a triangle and $[r]$ denotes the grea...
JEE Main 2024 (Online) 1st February Morning Shift
Let $3,7,11,15, \ldots, 403$ and $2,5,8,11, \ldots, 404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ______...
JEE Main 2024 (Online) 30th January Evening Shift
Let $$S_n$$ be the sum to $$n$$-terms of an arithmetic progression $$3,7,11$$, If $$40...
JEE Main 2024 (Online) 30th January Morning Shift
Let $$\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots$$ upto 10 terms and $$\beta=\sum_\limits{n=1}^{10} n^4$$. If $$4 \alpha-\beta=55 k+40$$, then $$\mathrm...
JEE Main 2024 (Online) 27th January Morning Shift
If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2 p)+\frac{1}{4^3}(3+3 p)+\cdots \cdots \infty$, then the value of $p$ is ____________.
JEE Main 2023 (Online) 15th April Morning Shift
If the sum of the series $\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{2^{2}}-\frac{1}{2 \cdot 3}+\frac{1}{3^{2}}\right)+\left(\frac{1}{2^{3}}...
JEE Main 2023 (Online) 13th April Morning Shift
The sum to $$20$$ terms of the series $$2 \cdot 2^{2}-3^{2}+2 \cdot 4^{2}-5^{2}+2 \cdot 6^{2}-\ldots \ldots$$ is equal to __________.
JEE Main 2023 (Online) 11th April Evening Shift
For $$k \in \mathbb{N}$$, if the sum of the series $$1+\frac{4}{k}+\frac{8}{k^{2}}+\frac{13}{k^{3}}+\frac{19}{k^{4}}+\ldots$$ is 10 , then the value o...
JEE Main 2023 (Online) 11th April Morning Shift
Let $$S=109+\frac{108}{5}+\frac{107}{5^{2}}+\ldots .+\frac{2}{5^{107}}+\frac{1}{5^{108}}$$. Then the value of $$\left(16 S-(25)^{-54}\right)$$ is equa...
JEE Main 2023 (Online) 10th April Evening Shift
Suppose $$a_{1}, a_{2}, 2, a_{3}, a_{4}$$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression i...
JEE Main 2023 (Online) 10th April Morning Shift
The sum of all those terms, of the arithmetic progression 3, 8, 13, ...., 373, which are not divisible by 3, is equal to ____________.
JEE Main 2023 (Online) 8th April Evening Shift
Let $$0 ...
JEE Main 2023 (Online) 6th April Evening Shift
If $$(20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17}+\ldots+20(21)^{19}=k(20)^{19}$$, then $$k$$ is equal to ___________.
JEE Main 2023 (Online) 1st February Evening Shift
The sum of the common terms of the following three arithmetic progressions. $$3,7,11,15, \ldots ., 399$$, $$2,5,8,11, \ldots ., 359$$ and $$2,7,12,17,...
JEE Main 2023 (Online) 1st February Morning Shift
Let $$a_{1}=8, a_{2}, a_{3}, \ldots, a_{n}$$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the ...
JEE Main 2023 (Online) 31st January Evening Shift
The sum $1^{2}-2 \cdot 3^{2}+3 \cdot 5^{2}-4 \cdot 7^{2}+5 \cdot 9^{2}-\ldots+15 \cdot 29^{2}$ is _________.
JEE Main 2023 (Online) 31st January Morning Shift
Let $$a_{1}, a_{2}, \ldots, a_{n}$$ be in A.P. If $$a_{5}=2 a_{7}$$ and $$a_{11}=18$$, then $$12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{...
JEE Main 2023 (Online) 30th January Evening Shift
The $8^{\text {th }}$ common term of the series $$ \begin{aligned} & S_1=3+7+11+15+19+\ldots . . \\\\ & S_2=1+6+11+16+21+\ldots . . \end{aligned} $$ i...
JEE Main 2023 (Online) 30th January Morning Shift
Let $$\sum_\limits{n=0}^{\infty} \frac{\mathrm{n}^{3}((2 \mathrm{n}) !)+(2 \mathrm{n}-1)(\mathrm{n} !)}{(\mathrm{n} !)((2 \mathrm{n}) !)}=\mathrm{ae}+...
JEE Main 2023 (Online) 29th January Evening Shift
Let $$a_1=b_1=1$$ and $${a_n} = {a_{n - 1}} + (n - 1),{b_n} = {b_{n - 1}} + {a_{n - 1}},\forall n \ge 2$$. If $$S = \sum\limits_{n = 1}^{10} {{{{b_n}}...
JEE Main 2023 (Online) 29th January Evening Shift
Let $$\{ {a_k}\} $$ and $$\{ {b_k}\} ,k \in N$$, be two G.P.s with common ratios $${r_1}$$ and $${r_2}$$ respectively such that $${a_1} = {b_1} = 4$$ ...
JEE Main 2023 (Online) 29th January Morning Shift
Let $$a_1,a_2,a_3,...$$ be a $$GP$$ of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh ter...
JEE Main 2023 (Online) 25th January Evening Shift
For the two positive numbers $$a,b,$$ if $$a,b$$ and $$\frac{1}{18}$$ are in a geometric progression, while $$\frac{1}{a},10$$ and $$\frac{1}{b}$$ are...
JEE Main 2023 (Online) 24th January Evening Shift
If $${{{1^3} + {2^3} + {3^3}\, + \,...\,up\,to\,n\,terms} \over {1\,.\,3 + 2\,.\,5 + 3\,.\,7\, + \,...\,up\,to\,n\,terms}} = {9 \over 5}$$, then the v...
JEE Main 2023 (Online) 24th January Morning Shift
The 4$$^\mathrm{th}$$ term of GP is 500 and its common ratio is $$\frac{1}{m},m\in\mathbb{N}$$. Let $$\mathrm{S_n}$$ denote the sum of the first n ter...
JEE Main 2022 (Online) 29th July Morning Shift
Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an A.P. If $$\sum\limits_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4$$, then $$4 a_{2}$$ is equal to _________.
JEE Main 2022 (Online) 29th July Morning Shift
If $$\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{100 \times 101 \times 102}=\frac{\mathr...
JEE Main 2022 (Online) 28th July Evening Shift
$${6 \over {{3^{12}}}} + {{10} \over {{3^{11}}}} + {{20} \over {{3^{10}}}} + {{40} \over {{3^9}}} + \,\,...\,\, + \,\,{{10240} \over 3} = {2^n}\,.\,m$...
JEE Main 2022 (Online) 27th July Evening Shift
$$ \frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30...
JEE Main 2022 (Online) 26th July Evening Shift
If $$\sum\limits_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}$$, where m and n are co-prime, then $$m+n$$ is equal to _____________.
JEE Main 2022 (Online) 26th July Evening Shift
Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all...
JEE Main 2022 (Online) 26th July Morning Shift
The series of positive multiples of 3 is divided into sets : $$\{3\},\{6,9,12\},\{15,18,21,24,27\}, \ldots$$ Then the sum of the elements in the $$11^...
JEE Main 2022 (Online) 25th July Morning Shift
Let $$a, b$$ be two non-zero real numbers. If $$p$$ and $$r$$ are the roots of the equation $$x^{2}-8 \mathrm{a} x+2 \mathrm{a}=0$$ and $$\mathrm{q}$$...
JEE Main 2022 (Online) 25th July Morning Shift
Let $$a_{1}=b_{1}=1, a_{n}=a_{n-1}+2$$ and $$b_{n}=a_{n}+b_{n-1}$$ for every natural number $$n \geqslant 2$$. Then $$\sum\limits_{n = 1}^{15} {{a_n}....
JEE Main 2022 (Online) 30th June Morning Shift
Let for $$f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1$$ and $$f'(1) = 0$$. If a0, a1, a2 are in an arithmatico-geometric progression, whose corresp...
JEE Main 2022 (Online) 29th June Evening Shift
Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is e...
JEE Main 2022 (Online) 28th June Evening Shift
Let for n = 1, 2, ......, 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose common ratio is $${1 \over {{{(n + ...
JEE Main 2022 (Online) 28th June Morning Shift
Let A = {1, a1, a2 ....... a18, 77} be a set of integers with 1 1 2 18 Let the set A + A = {x + y : x, y $$\in$$ A} contain exactly 39 elements. Then,...
JEE Main 2022 (Online) 27th June Morning Shift
If the sum of the first ten terms of the series $${1 \over 5} + {2 \over {65}} + {3 \over {325}} + {4 \over {1025}} + {5 \over {2501}} + \,\,....$$ is...
JEE Main 2022 (Online) 26th June Evening Shift
If a1 (> 0), a2, a3, a4, a5 are in a G.P., a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4, then a2 + a4 + 2a5 is equal to ___________....
JEE Main 2022 (Online) 25th June Morning Shift
For a natural number n, let $${\alpha _n} = {19^n} - {12^n}$$. Then, the value of $${{31{\alpha _9} - {\alpha _{10}}} \over {57{\alpha _8}}}$$ is ____...
JEE Main 2022 (Online) 25th June Morning Shift
The greatest integer less than or equal to the sum of first 100 terms of the sequence $${1 \over 3},{5 \over 9},{{19} \over {27}},{{65} \over {81}},$$...
JEE Main 2021 (Online) 31st August Evening Shift
The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____________.
JEE Main 2021 (Online) 31st August Evening Shift
If $$S = {7 \over 5} + {9 \over {{5^2}}} + {{13} \over {{5^3}}} + {{19} \over {{5^4}}} + ....$$, then 160 S is equal to ________.
JEE Main 2021 (Online) 26th August Evening Shift
The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is ____________...
JEE Main 2021 (Online) 26th August Evening Shift
Let a1, a2, ......., a10 be an AP with common difference $$-$$ 3 and b1, b2, ........., b10 be a GP with common ratio 2. Let ck = ak + bk, k = 1, 2, ....
JEE Main 2021 (Online) 27th July Morning Shift
If $${\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right)$$ are in an arithmetic progression, then the value of x is equal to...
JEE Main 2021 (Online) 25th July Morning Shift
If the value of $${\left( {1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + ....upto\,\infty } \right)^{{{\log }_{(0.25)}}\left( {{1 \over...
JEE Main 2021 (Online) 22th July Evening Shift
The sum of all the elements in the set {n$$\in$$ {1, 2, ....., 100} | H.C.F. of n and 2040 is 1} is equal to _____________.
JEE Main 2021 (Online) 20th July Evening Shift
For k $$\in$$ N, let $${1 \over {\alpha (\alpha + 1)(\alpha + 2).........(\alpha + 20)}} = \sum\limits_{K = 0}^{20} {{{{A_k}} \over {\alpha + k}}}...
JEE Main 2021 (Online) 20th July Evening Shift
Let $$\left\{ {{a_n}} \right\}_{n = 1}^\infty $$ be a sequence such that a1 = 1, a2 = 1 and $${a_{n + 2}} = 2{a_{n + 1}} + {a_n}$$ for all n $$\ge$$ 1...
JEE Main 2021 (Online) 16th March Evening Shift
Sn(x) = loga1/2x + loga1/3x + loga1/6x + loga1/11x + loga1/18x + loga1/27x + ...... up to n-terms, where a > 1. If S24(x) = 1093 and S12(2x) = 265,...
JEE Main 2021 (Online) 16th March Evening Shift
Let $${1 \over {16}}$$, a and b be in G.P. and $${1 \over a}$$, $${1 \over b}$$, 6 be in A.P., where a, b > 0. Then 72(a + b) is equal to _________...
JEE Main 2021 (Online) 16th March Morning Shift
Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these ser...
JEE Main 2021 (Online) 26th February Evening Shift
The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is _________.
JEE Main 2021 (Online) 26th February Evening Shift
If the arithmetic mean and geometric mean of the pth and qth terms of the sequence $$-$$16, 8, $$-$$4, 2, ...... satisfy the equation 4x2 $$-$$ 9x + 5...
JEE Main 2021 (Online) 25th February Morning Shift
Let A1, A2, A3, ....... be squares such that for each n $$ \ge $$ 1, the length of the side of An equals the length of diagonal of An+1. If the length...
JEE Main 2021 (Online) 24th February Evening Shift
The sum of first four terms of a geometric progression (G. P.) is $${{65} \over {12}}$$ and the sum of their respective reciprocals is $${{65} \over {...
JEE Main 2020 (Online) 3rd September Evening Slot
If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4th A.M. is equal to 2nd G.M., then m is equal ...
JEE Main 2020 (Online) 3rd September Morning Slot
The value of $${\left( {0.16} \right)^{{{\log }_{2.5}}\left( {{1 \over 3} + {1 \over {{3^2}}} + ....to\,\infty } \right)}}$$ is equal to ______.
JEE Main 2020 (Online) 9th January Evening Slot
The number of terms common to the two A.P.'s 3, 7, 11, ....., 407 and 2, 9, 16, ....., 709 is ______.
JEE Main 2020 (Online) 8th January Evening Slot
The sum, $$\sum\limits_{n = 1}^7 {{{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 4}} $$ is equal to ________.
JEE Main 2020 (Online) 8th January Morning Slot
The sum $$\sum\limits_{k = 1}^{20} {\left( {1 + 2 + 3 + ... + k} \right)} $$ is :
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