Mathematical Induction and Binomial Theorem · Mathematics · JEE Advanced

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^5$ in the expansion of $\left(a x^2+\frac{70}{27 b x}\right)^4$ is equal to the ...

Let $$X = {({}^{10}{C_1})^2} + 2{({}^{10}{C_2})^2} + 3{({}^{10}{C_3})^2} + ... + 10{({}^{10}{C_{10}})^2}$$, where $${}^{10}{C_r}$$, r $$ \in $${1, 2, ...

Let $$m$$ be the smallest positive integer such that the coefficient of $${x^2}$$ in the expansion of $${\left( {1 + x} \right)^2} + {\left( {1 + x} \...

The coefficient of three consecutive terms of $${\left( {1 + x} \right)^{n + 5}}$$ are in the ratio $$5:10:14.$$ Then $$n$$ =

For non-negative integers s and r, let$$\left( {\matrix{ s \cr r \cr } } \right) = \left\{ {\matrix{ {{{s!} \over {r!(s - r)!}}} & ...

Coefficient of $${x^{11}}$$ in the expansion of $${\left( {1 + {x^2}} \right)^4}{\left( {1 + {x^3}} \right)^7}{\left( {1 + {x^4}} \right)^{12}}$$ is

For $$r = 0,\,1,....,$$ let $${A_r},\,{B_r}$$ and $${C_r}$$ denote, respectively, the coefficient of $${X^r}$$ in the expansions of $${\left( {1 + x}...

The value of $$$\left( {\matrix{ {30} \cr 0 \cr } } \right)\left( {\matrix{ {30} \cr {10} \cr } } \right) - \left( {\matrix{ ...

If $${}^{n - 1}{C_r} = \left( {{k^2} - 3} \right)\,{}^n{C_{r + 1,}}$$ then $$k \in $$

Coefficient of $${t^{24}}$$ in $${\left( {1 + {t^2}} \right)^{12}}\left( {1 + {t^{12}}} \right)\left( {1 + {t^{24}}} \right)$$ is

The sum $$\sum\limits_{i = 0}^m {\left( {\matrix{ {10} \cr i \cr } } \right)\left( {\matrix{ {20} \cr {m - i} \cr } } \right),...

In the binomial expansion of $${\left( {a - b} \right)^n},\,n \ge 5,$$ the sum of the $${5^{th}}$$ and $${6^{th}}$$ terms is zero. Then $$a/b$$ equals...

For $$2 \le r \le n,\,\,\,\,\left( {\matrix{ n \cr r \cr } } \right) + 2\left( {\matrix{ n \cr {r - 1} \cr } } \right) + \left...

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the coefficients of $$x$$ and $${x^2}$$ are $$3$$ and $$-6$$ respecti...

If $${a_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}},\,\,\,then\,\,\,\sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}}} } $$ equals

The expansion $${\left( {x + {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \ri...

If $${C_r}$$ stands for $${}^n{C_r},$$ then the sum of the series $${{2\left( {{n \over 2}} \right){\mkern 1mu} !{\mkern 1mu} \left( {{n \over 2}} \...

Given positive integers $$r > 1,\,n > 2$$ and that the coefficient of $$\left( {3r} \right)$$th and $$\left( {r + 2} \right)$$th terms in the bi...

The coefficient of $${x^4}$$ in $${\left( {{x \over 2} - {3 \over {{x^2}}}} \right)^{10}}$$ is

Prove that $${2^k}\left( {\matrix{ n \cr 0 \cr } } \right)\left( {\matrix{ n \cr k \cr } } \right) - {2^{^{k - 1}\left( {\mat...

Use mathematical induction to show that $${\left( {25} \right)^{n + 1}} - 24n + 5735$$ is divisible by $${\left( {24} \right)^2}$$ for all $$ = n = 1...

For every possitive integer $$n$$, prove that $$\sqrt {\left( {4n + 1} \right)} < \sqrt n + \sqrt {n + 1} < \sqrt {4n + 2}.$$ Hence or othe...

For any positive integer $$m$$, $$n$$ (with $$n \ge m$$), let $$\left( {\matrix{ n \cr m \cr } } \right) = {}^n{C_m}$$ Prove that $$\le...

Let $$a,\,b,\,c$$ be possitive real numbers such that $${b^2} - 4ac > 0$$ and let $${\alpha _1} = c.$$ Prove by induction that $${\alpha _{n + 1}} ...

A coin probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecutive...

Let $$n$$ be any positive integer. Prove that $$$\sum\limits_{k = 0}^m {{{\left( {\matrix{ {2n - k} \cr k \cr } } \right)} \over {\left( ...

Let $$p$$ be a prime and $$m$$ a positive integer. By mathematical induction on $$m$$, or otherwise, prove that whenever $$r$$ is an integer such that...

Let $$0 < {A_i} < n$$ for $$i = 1,\,2....,\,n.$$ Use mathematical induction to prove that $$$\sin {A_1} + \sin {A_2}....... + \sin {A_n} \le n\...

Using mathematical induction prove that for every integer $$n \ge 1,\,\,\left( {{3^{2n}} - 1} \right)$$ is divisible by $${2^{n + 2}}$$ but not by $${...

Let $$n$$ be a positive integer and $${\left( {1 + x + {x^2}} \right)^n} = {a_0} + {a_1}x + ............ + {a_{2n}}{x^{2n}}$$ Show that $$a_0^2 - a_1...

If $$x$$ is not an integral multiple of $$2\pi $$ use mathematical induction to prove that : $$$\cos x + \cos 2x + .......... + \cos nx = \cos {{n + ...

Using mathematical induction, prove that $${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1...

Prove that $$\sum\limits_{r = 1}^k {{{\left( { - 3} \right)}^{r - 1}}\,\,{}^{3n}{C_{2r - 1}} = 0,} $$ where $$k = \left( {3n} \right)/2$$ and $$n$$ is...

Let $$p \ge 3$$ be an integer and $$\alpha $$, $$\beta $$ be the roots of $${x^2} - \left( {p + 1} \right)x + 1 = 0$$ using mathematical induction sho...

If $$\sum\limits_{r = 0}^{2n} {{a_r}{{\left( {x - 2} \right)}^r}\,\, = \sum\limits_{r = 0}^{2n} {{b_r}{{\left( {x - 3} \right)}^r}} } $$ and $${a_k} =...

Using induction or otherwise, prove that for any non-negative integers $$m$$, $$n$$, $$r$$ and $$k$$ , $$\sum\limits_{m = 0}^k {\left( {n - m} \right)...

Prove that $${{{n^7}} \over 7} + {{{n^5}} \over 5} + {{2{n^3}} \over 3} - {n \over {105}}$$ is an integer for every positive integer $$n$$

Prove that $${C_0} - {2^2}{C_1} + {3^2}{C_2}\,\, - \,..... + {\left( { - 1} \right)^n}{\left( {n + 1} \right)^2}{C_n} = 0,\,\,\,\,n > 2,\,\,$$ wh...

Using mathematical induction, prove that $${}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}...

Let $$R$$ $$ = {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}$$ and $$f = R - \left[ R \right],$$ where [ ] denotes the greatest integer function. Prove t...

Prove by mathematical induction that $$ - 5 - {{\left( {2n} \right)!} \over {{2^{2n}}{{\left( {n!} \right)}^2}}} \le {1 \over {{{\left( {3n + 1} \righ...

Use method of mathematical induction $${2.7^n} + {3.5^n} - 5$$ is divisible by $$24$$ for all $$n > 0$$

Given $${s_n} = 1 + q + {q^2} + ...... + {q^2};$$ $${S_n} = 1 + {{q + 1} \over 2} + {\left( {{{q + 1} \over 2}} \right)^2} + ........ + {\left( {{{q ...

If $$p$$ be a natural number then prove that $${p^{n + 1}} + {\left( {p + 1} \right)^{2n - 1}}$$ is divisible by $${p^2} + p + 1$$ for every positive ...

Use mathematical Induction to prove : If $$n$$ is any odd positive integer, then $$n\left( {{n^2} - 1} \right)$$ is divisible by 24.

If $${\left( {1 + x} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ..... + {C_n}{x^n}$$ then show that the sum of the products of the $${C_i}s$$ taken t...

Prove that $${7^{2n}} + \left( {{2^{3n - 3}}} \right)\left( {3n - 1} \right)$$ is divisible by 25 for any natural number $$n$$.

Given that $${C_1} + 2{C_2}x + 3{C_3}{x^2} + ......... + 2n{C_{2n}}{x^{2n - 1}} = 2n{\left( {1 + x} \right)^{2n - 1}}$$ where $${C_r} = {{\left( {2n}...

The sum of the rational terms in the expansion of $${\left( {\sqrt 2 + {3^{1/5}}} \right)^{10}}$$ is ...............

Let $$n$$ be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of $${\left( {1 + x} \right)^n}$$ are in A.P., then the...

If $${\left( {1 + ax} \right)^n} = 1 + 8x + 24{x^2} + .....$$ then $$a=..........$$ and $$n =............$$

The sum of the coefficients of the plynomial $${\left( {1 + x - 3{x^2}} \right)^{2163}}$$ is ...............

The larger of $${99^{50}} + {100^{50}}$$ and $${101^{50}}$$ is ..............