Mathematics
Mathematical Induction and Binomial Theorem
Previous Years Questions

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

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 ## Subjective 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 any positive integer $$m$$, $$n$$ (with $$n \ge m$$), let $$\left( {\matrix{ n \cr m \cr } } \right) = {}^n{C_m}$$ Prove that $$\le... For every possitive integer$$n$$, prove that$$\sqrt {\left( {4n + 1} \right)} < \sqrt n + \sqrt {n + 1} < \sqrt {4n + 2}.$$Hence or othe... Let$$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 $${... If$$x$$is not an integral multiple of$$2\pi $$use mathematical induction to prove that :$$\$\cos x + \cos 2x + .......... + \cos nx = \cos {{n + ...
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 $$a_0^2 - a_1... Prove that$$\sum\limits_{r = 1}^k {{{\left( { - 3} \right)}^{r - 1}}\,\,{}^{3n}{C_{2r - 1}} = 0,} $$where$$k = \left( {3n} \right)/2$$and$$n$$is... Using mathematical induction, prove that$${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1...
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 $${a_k} =... Let$$p \ge 3$$be an integer and$$\alpha $$,$$\beta $$be the roots of$${x^2} - \left( {p + 1} \right)x + 1 = 0$$using mathematical induction sho... Using induction or otherwise, prove that for any non-negative integers$$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 $$n$$
Using mathematical induction, prove that $${}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}... Prove that$${C_0} - {2^2}{C_1} + {3^2}{C_2}\,\, - \,..... + {\left( { - 1} \right)^n}{\left( {n + 1} \right)^2}{C_n} = 0,\,\,\,\,n > 2,\,\,$$wh... Let$$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 $$n > 0$$
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 ...
Given $${s_n} = 1 + q + {q^2} + ...... + {q^2};$$ $${S_n} = 1 + {{q + 1} \over 2} + {\left( {{{q + 1} \over 2}} \right)^2} + ........ + {\left( {{{q ... If$${\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$$taken t... Use mathematical Induction to prove : If$$n$$is any odd positive integer, then$$n\left( {{n^2} - 1} \right)$$is divisible by 24. Prove that$${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}...

## Fill in the Blanks

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 larger of $${99^{50}} + {100^{50}}$$ and $${101^{50}}$$ is ..............
The sum of the coefficients of the plynomial $${\left( {1 + x - 3{x^2}} \right)^{2163}}$$ is ...............
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