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## Sequences and Series

### Mathematics

#### Previous Years Questions

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\begin{aligned} &\text { Let }\left\{a_{n}\right\}_{n=0}^{\infty} \text { be a sequence such that } a_{0}=a_{1}=0 \te... JEE Main 2022 (Online) 29th July Evening Shift Consider the sequencea_{1}, a_{2}, a_{3}, \ldots$$such that$$a_{1}=1, a_{2}=2$$and$$a_{n+2}=\frac{2}{a_{n+1}}+a_{...
JEE Main 2022 (Online) 28th July Morning 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 t...
JEE Main 2022 (Online) 27th July Evening Shift
Suppose $$a_{1}, a_{2}, \ldots, a_{n}$$, .. be an arithmetic progression of natural numbers. If the ratio of the sum of ...
JEE Main 2022 (Online) 27th 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...
JEE Main 2022 (Online) 26th July Morning Shift
The sum $$\sum\limits_{n = 1}^{21} {{3 \over {(4n - 1)(4n + 3)}}}$$ is equal to
JEE Main 2022 (Online) 25th July Evening Shift
The value of $$1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + ... JEE Main 2022 (Online) 30th June Morning Shift If$$\{ {a_i}\} _{i = 1}^n$$, where n is an even integer, is an arithmetic progression with common difference 1, and$$\...
JEE Main 2022 (Online) 24th June Morning Shift
Let x, y > 0. If x3y2 = 215, then the least value of 3x + 2y is
JEE Main 2022 (Online) 24th June Evening Shift
If $${b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N}$$, then
JEE Main 2022 (Online) 25th June Evening Shift
The sum 1 + 2 . 3 + 3 . 32 + ......... + 10 . 39 is equal to :
JEE Main 2022 (Online) 25th June Evening Shift
If $$A = \sum\limits_{n = 1}^\infty {{1 \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}}$$ and $$B = \sum\limits_{n =... JEE Main 2022 (Online) 26th June Evening Shift$$x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} ...
JEE Main 2022 (Online) 27th June Morning 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 Evening Shift
Let $$S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} + \,.....$$. Then 4S is e...
JEE Main 2022 (Online) 27th June Evening Shift
Let A1, A2, A3, ....... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = $${1 \over {1296}... JEE Main 2022 (Online) 28th June Morning Shift If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and... JEE Main 2022 (Online) 28th June Evening Shift The sum of the infinite series$$1 + {5 \over 6} + {{12} \over {{6^2}}} + {{22} \over {{6^3}}} + {{35} \over {{6^4}}} + ...
JEE Main 2022 (Online) 29th June Evening Shift
Let $$\{ {a_n}\} _{n = 0}^\infty$$ be a sequence such that $${a_0} = {a_1} = 0$$ and $${a_{n + 2}} = 2{a_{n + 1}} - {a_... JEE Main 2022 (Online) 29th June Morning Shift Let a1, a2, ..........., a21 be an AP such that$$\sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}} ...
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 $$\sum\limit... JEE Main 2021 (Online) 1st September Evening Shift Let a1, a2, a3, ..... be an A.P. If$${{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} ...
JEE Main 2021 (Online) 31st August Evening Shift
Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the ...
JEE Main 2021 (Online) 31st August Morning Shift
The sum of 10 terms of the series $${3 \over {{1^2} \times {2^2}}} + {5 \over {{2^2} \times {3^2}}} + {7 \over {{3^2} \... JEE Main 2021 (Online) 31st August Morning Shift If 0 &lt; x &lt; 1 and$$y = {1 \over 2}{x^2} + {2 \over 3}{x^3} + {3 \over 4}{x^4} + ....$$, then the value of e1 + y a... JEE Main 2021 (Online) 27th August Evening Shift If for x, y$$\in$$R, x &gt; 0, y = log10x + log10x1/3 + log10x1/9 + ...... upto$$\infty$$terms and$${{2 + 4 + 6 + ....
JEE Main 2021 (Online) 27th August Morning Shift
If 0 &lt; x &lt; 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 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...
JEE Main 2021 (Online) 26th August Morning Shift
The sum of the series $${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} ... JEE Main 2021 (Online) 26th August Morning Shift Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of$${{{S_{4n}}} \ove...
JEE Main 2021 (Online) 25th July Morning Shift
Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 $$-$$ S6 is equal ...
JEE Main 2021 (Online) 22th July Evening Shift
If sum of the first 21 terms of the series $${\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + ......... JEE Main 2021 (Online) 20th July Evening Shift Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithm... JEE Main 2021 (Online) 18th March Evening Shift$${1 \over {{3^2} - 1}} + {1 \over {{5^2} - 1}} + {1 \over {{7^2} - 1}} + .... + {1 \over {{{(201)}^2} - 1}}$$is equal ... JEE Main 2021 (Online) 18th March Morning Shift If$$\alpha$$,$$\beta$$are natural numbers such that 100$$\alpha-$$199$$\beta$$= (100)(100) + (99)(101) + (98)(... JEE Main 2021 (Online) 18th March Morning Shift The sum of the series$$\sum\limits_{n = 1}^\infty {{{{n^2} + 6n + 10} \over {(2n + 1)!}}} $$is equal to : JEE Main 2021 (Online) 26th February Evening Shift In an increasing geometric series, the sum of the second and the sixth term is$${{25} \over 2}$$and the product of the... JEE Main 2021 (Online) 26th February Morning Shift The sum of the infinite series$$1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{2...
JEE Main 2021 (Online) 26th February Morning Shift
If $$0 &lt; \theta ,\phi &lt; {\pi \over 2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } ,y = \sum\limits_{... JEE Main 2021 (Online) 25th February Morning Slot The common difference of the A.P. b1, b2, … , bm is 2 more than the common difference of A.P. a1, a2, …, an. If a40 = ... JEE Main 2020 (Online) 6th September Evening Slot Let a , b, c , d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + ... JEE Main 2020 (Online) 6th September Morning Slot If the sum of the first 20 terms of the series$${\log _{\left( {{7^{1/2}}} \right)}}x + {\log _{\left( {{7^{1/3}}} \rig...
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 eigh...
JEE Main 2020 (Online) 5th September Evening 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 $$\al... JEE Main 2020 (Online) 5th September Morning Slot If 210 + 29.31 + 28 .32 +.....+ 2.39 + 310 = S - 211, then S is equal to :... JEE Main 2020 (Online) 5th September Morning Slot Let a1, a2, ..., an be a given A.P. whose common difference is an integer and Sn = a1 + a2 + .... + an. If a1 = 1, an = ... JEE Main 2020 (Online) 4th September Evening Slot If 1+(1–22.1)+(1–42.3)+(1-62.5)+......+(1-202.19)=$$\alpha $$- 220$$\beta $$, then an ordered pair$$\left( {\alpha ,...
JEE Main 2020 (Online) 4th September Morning 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 t...
JEE Main 2020 (Online) 3rd September Evening 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 c...
JEE Main 2020 (Online) 3rd September Morning 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 + (...
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 ,...
JEE Main 2020 (Online) 2nd September Evening 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) 2nd September Morning Slot
If |x| &lt; 1, |y| &lt; 1 and x $$\ne$$ y, then the sum to infinity of the following series (x + y) + (x2+xy+y2) + (x3...
JEE Main 2020 (Online) 2nd September Morning Slot
Let an be the nth term of a G.P. of positive terms. $$\sum\limits_{n = 1}^{100} {{a_{2n + 1}} = 200}$$ and $$\sum\limit... JEE Main 2020 (Online) 9th January Evening Slot The product$${2^{{1 \over 4}}}{.4^{{1 \over {16}}}}{.8^{{1 \over {48}}}}{.16^{{1 \over {128}}}}$$... to$$\infty $$is... JEE Main 2020 (Online) 9th January Morning Slot If the 10th term of an A.P. is$${1 \over {20}}$$and its 20th term is$${1 \over {10}}$$, then the sum of its first 200... JEE Main 2020 (Online) 8th January Evening Slot Let ƒ : R$$ \to $$R be such that for all x$$ \in $$R (21+x + 21–x), ƒ(x) and (3x + 3–x) are in A.P., then the minim... JEE Main 2020 (Online) 8th January Morning Slot If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to ... JEE Main 2020 (Online) 7th January Evening Slot Let$${a_1}$$,$${a_2}$$,$${a_3}$$,....... be a G.P. such that$${a_1}$$&lt; 0,$${a_1}$$+$${a_2}$$= 4 and$$...
JEE Main 2020 (Online) 7th January Evening 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 th...
JEE Main 2020 (Online) 7th January Morning 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 Evening 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) 12th April Morning Slot
For x $$\varepsilon$$ R, let [x] denote the greatest integer $$\le$$ x, then the sum of the series $$\left[ { - {1 \o... JEE Main 2019 (Online) 12th April Morning Slot Let a1, a2, a3,......be an A.P. with a6 = 2. Then the common difference of this A.P., which maximises the product a1a4a5... JEE Main 2019 (Online) 10th April Evening Slot Let$$a$$, b and c be in G.P. with common ratio r, where$$a \ne $$0 and 0 &lt; r$$ \le {1 \over 2}$$. If 3... JEE Main 2019 (Online) 10th April Evening Slot The sum$$1 + {{{1^3} + {2^3}} \over {1 + 2}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 2 + 3}} + ...... + {{{1^3} + {2^3} +...
JEE Main 2019 (Online) 10th April Evening Slot
If a1, a2, a3, ............... an are in A.P. and a1 + a4 + a7 + ........... + a16 = 114, then a1 + a6 + a11 + a16 is eq...
JEE Main 2019 (Online) 10th April Morning Slot
The sum $${{3 \times {1^3}} \over {{1^3}}} + {{5 \times ({1^3} + {2^3})} \over {{1^2} + {2^2}}} + {{7 \times \left( {{1^... JEE Main 2019 (Online) 10th April Morning Slot The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 +.... upto 11th term is :- JEE Main 2019 (Online) 9th April Evening Slot If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its 11th term i... JEE Main 2019 (Online) 9th April Evening Slot Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the secon... JEE Main 2019 (Online) 9th April Evening Slot Let the sum of the first n terms of a non-constant A.P., a1, a2, a3, ..... be$$50n + {{n(n - 7)} \over 2}A$$, where A i... JEE Main 2019 (Online) 9th April Morning Slot If three distinct numbers a, b, c are in G.P. and the equations ax2 + 2bx + c = 0 and dx2 + 2ex + ƒ = 0 have a common ... JEE Main 2019 (Online) 8th April Evening Slot The sum$$\sum\limits_{k = 1}^{20} {k{1 \over {{2^k}}}} $$is equal to JEE Main 2019 (Online) 8th April Evening Slot The sum of all natural numbers 'n' such that 100 &lt; n &lt; 200 and H.C.F. (91, n) &gt; 1 is : JEE Main 2019 (Online) 8th April Morning Slot If &nbsp;&nbsp;nC4, nC5 and nC6 are in A.P., then n can be :... JEE Main 2019 (Online) 12th January Evening Slot If sin4$$\alpha $$+ 4 cos4$$\beta $$+ 2 = 4$$\sqrt 2 $$sin$$\alpha $$cos$$\beta $$;$$\alpha $$,$$\beta  \in...
JEE Main 2019 (Online) 12th January Evening Slot
If the sum of the first 15 terms of the series $${\left( {{3 \over 4}} \right)^3} + {\left( {1{1 \over 2}} \right)^3} + ... JEE Main 2019 (Online) 12th January Evening Slot The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these term... JEE Main 2019 (Online) 12th January Morning Slot Let&nbsp;&nbsp;Sk =$${{1 + 2 + 3 + .... + k} \over k}.$$If &nbsp;&nbsp;$$S_1^2 + S_2^2 + .....\, + S_{10}^2 = {5 \over...
JEE Main 2019 (Online) 12th January Morning Slot
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $${{{x^m}{y^n}} \over ... JEE Main 2019 (Online) 11th January Evening Slot If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is : JEE Main 2019 (Online) 11th January Evening Slot Let a1, a2, . . . . . ., a10 be a G.P. &nbsp;&nbsp; If$${{{a_3}} \over {{a_1}}} = 25,$$then$${{{a_9}} \over {{a_5...
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 $${{27} \over ... JEE Main 2019 (Online) 11th January Morning Slot Let a1, a2, a3, ..... a10 be in G.P. with ai &gt; 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k$$ \in ...
JEE Main 2019 (Online) 10th January Evening 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) 10th January Morning Slot
Let $${a_1},{a_2},.......,{a_{30}}$$ be an A.P., $$S = \sum\limits_{i = 1}^{30} {{a_i}}$$ and $$T = \sum\limits_{i = 1}... JEE Main 2019 (Online) 9th January Morning Slot Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also three consecutive ... JEE Main 2019 (Online) 9th January Evening Slot The sum of the following series$$1 + 6 + {{9\left( {{1^2} + {2^2} + {3^2}} \right)} \over 7} + {{12\left( {{1^2} + {2^2...
JEE Main 2019 (Online) 9th January Evening 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 $${1 \over {{x_1}}},{1 \over {{x_2}}},...,{1 \over {{x_n}}}\,\,$$ (xi $$\ne$$ 0 for i = 1, 2, ..., n) be in A.P. s...
JEE Main 2018 (Online) 16th April Morning Slot
The sum of the first 20 terms of the series $$1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ...,... JEE Main 2018 (Online) 16th April Morning Slot Let &nbsp;&nbsp; An =$$\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^...
JEE Main 2018 (Online) 15th April Evening Slot
If &nbsp;a, &nbsp; b, &nbsp; c&nbsp; are in A.P. and&nbsp; a2, &nbsp;b2, &nbsp;c2 are in G.P. such that a&nbsp;&lt;&nbs...
JEE Main 2018 (Online) 15th April Evening Slot
If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :
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...
JEE Main 2018 (Online) 15th April Morning Slot
Let $${a_1}$$, $${a_2}$$, $${a_3}$$, ......... ,$${a_{49}}$$ be in A.P. such that $$\sum\limits_{k = 0}^{12} {{a_{4k + 1... JEE Main 2018 (Offline) Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 12 + 2.22 + 32 + 2.42 + 52 +... JEE Main 2018 (Offline) Let Sn =$${1 \over {{1^3}}} + {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + .......
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
If the sum of the first n terms of the series $$\,\sqrt 3 + \sqrt {75} + \sqrt {243} + \sqrt {507} + ......$$ is $$... JEE Main 2017 (Online) 8th April Morning Slot If the arithmetic mean of two numbers a and b, a &gt; b &gt; 0, is five times their geometric mean, then$${{a + b} \ov...
JEE Main 2017 (Online) 8th April Morning Slot
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 2017 (Offline)
Let $$a$$, b, c $$\in R$$. If $$f$$(x) = ax2 + bx + c is such that $$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f... JEE Main 2017 (Offline) If&nbsp;&nbsp; A &gt; 0, B &gt; 0&nbsp;&nbsp; and &nbsp;&nbsp; A + B =$${\pi \over 6}$$, then the minimum value of t... JEE Main 2016 (Online) 10th April Morning Slot Let a1, a2, a3, . . . . . . . , an, . . . . . be in A.P. If a3 + a7 + a11 + a15 = 72, then the sum of its first 17 te... JEE Main 2016 (Online) 10th April Morning Slot Let z = 1 + ai be a complex number, a &gt; 0, such that z3 is a real number. Then the sum 1 + z + z2 + . . . . .+ z11 ... JEE Main 2016 (Online) 10th April Morning Slot Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3is equal to :... JEE Main 2016 (Online) 9th April Morning Slot For x$$ \in $$R, x$$ \ne $$-1, if (1 + x)2016 + x(1 + x)2015 + x2(1 + x)2014 + . . . . + x2016 =$$\sum\limits_{i ...
JEE Main 2016 (Online) 9th April Morning Slot
If the sum of the first ten terms of the series $${\left( {1{3 \over 5}} \right)^2} + {\left( {2{2 \over 5}} \right)^2} ... JEE Main 2016 (Offline) If the$${2^{nd}},{5^{th}}\,and\,{9^{th}}$$terms of a non-constant A.P. are in G.P., then the common ratio of this G.P.... JEE Main 2016 (Offline) If m is the A.M. of two distinct real numbers l and n$$(l,n &gt; 1)$$and$${G_1},{G_2}$$and$${G_3}$$are three geome... JEE Main 2015 (Offline) The sum of first 9 terms of the series.$${{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}...
JEE Main 2015 (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 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. t...
JEE Main 2014 (Offline)
The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is
JEE Main 2013 (Offline)
Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000. ...
AIEEE 2012
A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increase...
AIEEE 2011
A person is to count 4500 currency notes. Let $${a_n}$$ denote the number of notes he counts in the $${n^{th}}$$ minute....
AIEEE 2010
The sum to infinite term of the series $$1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}... AIEEE 2009 The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the ter... AIEEE 2008 In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common... AIEEE 2007 The sum of series$${1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......$$upto infinity is AIEEE 2007 Let$${a_1}$$,$${a_2}$$,$${a_3}$$.....be terms on A.P. If$${{{a_1} + {a_2} + .....{a_p}} \over {{a_1} + {a_2} + ........
AIEEE 2006
If $${{a_1},{a_2},....{a_n}}$$ are in H.P., then the expression $${{a_1}\,{a_2} + \,{a_2}\,{a_3}\, + .... + {a_{n - 1}}\... AIEEE 2006 The sum of the series$$1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + .......$$ad inf. is AIEEE 2005 If$$x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\inft...
AIEEE 2005
The sum of series $${1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........$$ is
AIEEE 2004
The sum of the first n terms of the series $${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1... AIEEE 2004 Let$${{T_r}}$$be the rth term of an A.P. whose first term is a and common difference is d. If for some positive intege... AIEEE 2004 The sum of the serier$${1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..............$$up to$$\infty $$is equal t... AIEEE 2003$${1^3} - \,\,{2^3} + {3^3} - {4^3} + ... + {9^3} = $$AIEEE 2002 Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is AIEEE 2002 Fifth term of a GP is 2, then the product of its 9 terms is AIEEE 2002 The value of$$\,{2^{1/4}}.\,\,{4^{1/8}}.\,{8^{1/16}}...\infty $$is AIEEE 2002 l, m, n are the$${p^{th}}$$,$${q^{th}}$$and$${r^{th}}$$term of a G.P all positive,$$then\,\left| {\matrix{ {\lo...
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

## Numerical

More
If $$\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{100 \time... JEE Main 2022 (Online) 29th July Morning Shift Let$$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}$$... JEE Main 2022 (Online) 29th July Morning Shift$${6 \over {{3^{12}}}} + {{10} \over {{3^{11}}}} + {{20} \over {{3^{10}}}} + {{40} \over {{3^9}}} + \,\,...\,\, + \,\,{{...
JEE Main 2022 (Online) 28th July Evening Shift
$$\frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}... JEE Main 2022 (Online) 27th July Evening Shift Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of... JEE Main 2022 (Online) 26th July Evening Shift If$$\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$$is equa... JEE Main 2022 (Online) 26th July Evening Shift The series of positive multiples of 3 is divided into sets :$$\{3\},\{6,9,12\},\{15,18,21,24,27\}, \ldots$$Then the su... JEE Main 2022 (Online) 26th July Morning Shift Let$$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$$. Then$$\s...
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 $$x^{2}-8 \mathrm{a} x+2 \ma... JEE Main 2022 (Online) 25th July Morning Shift The greatest integer less than or equal to the sum of first 100 terms of the sequence$${1 \over 3},{5 \over 9},{{19} \o...
JEE Main 2022 (Online) 25th June Morning Shift
For a natural number n, let $${\alpha _n} = {19^n} - {12^n}$$. Then, the value of $${{31{\alpha _9} - {\alpha _{10}}} \o... JEE Main 2022 (Online) 25th June Morning Shift If a1 (> 0), a2, a3, a4, a5 are in a G.P., a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4, then a2 + a4 + 2a5 is equal to ________... JEE Main 2022 (Online) 26th June Evening Shift If the sum of the first ten terms of the series$${1 \over 5} + {2 \over {65}} + {3 \over {325}} + {4 \over {1025}} + {5...
JEE Main 2022 (Online) 27th June Morning Shift
Let for n = 1, 2, ......, 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose commo...
JEE Main 2022 (Online) 28th 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 ...
JEE Main 2022 (Online) 29th June 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) 31st August Evening Shift
The mean of 10 numbers 7 $$\times$$ 8, 10 $$\times$$ 10, 13 $$\times$$ 12, 16 $$\times$$ 14, ....... is ____________.
JEE Main 2021 (Online) 31st August Morning Shift
Let a1, a2, ......., a10 be an AP with common difference $$-$$ 3 and b1, b2, ........., b10 be a GP with common ratio 2....
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 m...
JEE Main 2021 (Online) 26th August Evening Shift
If $${\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right)$$ are in an arithmetic progression, t...
JEE Main 2021 (Online) 27th July Morning Shift
If the value of $${\left( {1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + ....upto\,\infty } \right)^{{{\l... JEE Main 2021 (Online) 25th July Morning Shift The sum of all the elements in the set {n$$\in$${1, 2, ....., 100} | H.C.F. of n and 2040 is 1} is equal to ___________... JEE Main 2021 (Online) 22th July Evening Shift Let$$\left\{ {{a_n}} \right\}_{n = 1}^\infty $$be a sequence such that a1 = 1, a2 = 1 and$${a_{n + 2}} = 2{a_{n + 1}}...
JEE Main 2021 (Online) 20th July Evening Shift
For k $$\in$$ N, let $${1 \over {\alpha (\alpha + 1)(\alpha + 2).........(\alpha + 20)}} = \sum\limits_{K = 0}^{20} {... JEE Main 2021 (Online) 20th July Evening Shift Let$${1 \over {16}}$$, a and b be in G.P. and$${1 \over a}$$,$${1 \over b}$$, 6 be in A.P., where a, b &gt; 0. Then 7... JEE Main 2021 (Online) 16th March Evening Shift Sn(x) = loga1/2x + loga1/3x + loga1/6x + loga1/11x + loga1/18x + loga1/27x + ...... up to n-terms, where a &gt; 1. If S2... JEE Main 2021 (Online) 16th March Evening Shift Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. ... JEE Main 2021 (Online) 16th March Morning Shift If the arithmetic mean and geometric mean of the pth and qth terms of the sequence$$-$$16, 8,$$-$$4, 2, ...... satisfy... JEE Main 2021 (Online) 26th February Evening Shift The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is _________. JEE Main 2021 (Online) 26th February Evening Shift Let A1, A2, A3, ....... be squares such that for each n$$ \ge $$1, the length of the side of An equals the length of d... JEE Main 2021 (Online) 25th February Morning Slot If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4th A.M. is equal... JEE Main 2020 (Online) 3rd September Evening Slot The value of$${\left( {0.16} \right)^{{{\log }_{2.5}}\left( {{1 \over 3} + {1 \over {{3^2}}} + ....to\,\infty } \right)...
JEE Main 2020 (Online) 3rd September Morning 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) 9th 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 Evening Slot
The sum $$\sum\limits_{k = 1}^{20} {\left( {1 + 2 + 3 + ... + k} \right)}$$ is :
JEE Main 2020 (Online) 8th January Morning Slot

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