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Mathematical Induction and Binomial Theorem

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2025 JEE Advanced Numerical
JEE Advanced 2025 Paper 2 Online

Let $a_0, a_1, \ldots, a_{23}$ be real numbers such that

$ \left(1+\frac{2}{5} x\right)^{23}=\sum\limits_{i=0}^{23} a_i x^i $

for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0 \leq j \leq 23$. Then the value of $r$ is ____________.

2023 JEE Advanced Numerical
JEE Advanced 2023 Paper 1 Online
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 coefficient of $x^{-5}$ in the expansion of $\left(a x-\frac{1}{b x^2}\right)^7$, then the value of $2 b$ is :
2018 JEE Advanced Numerical
JEE Advanced 2018 Paper 2 Offline
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, ..., 10} denote binomial coefficients. Then, the value of ${1 \over {1430}}X$ is ..........
2016 JEE Advanced Numerical
JEE Advanced 2016 Paper 1 Offline
Let $m$ be the smallest positive integer such that the coefficient of ${x^2}$ in the expansion of ${\left( {1 + x} \right)^2} + {\left( {1 + x} \right)^3} + ........ + {\left( {1 + x} \right)^{49}} + {\left( {1 + mx} \right)^{50}}\,\,$ is $\left( {3n + 1} \right)\,{}^{51}{C_3}$ for some positive integer $n$. Then the value of $n$ is
2013 JEE Advanced Numerical
JEE Advanced 2013 Paper 1 Offline
The coefficient of three consecutive terms of ${\left( {1 + x} \right)^{n + 5}}$ are in the ratio $5:10:14.$ Then $n$ =
2003 JEE Advanced Numerical
IIT-JEE 2003
Prove that
${2^k}\left( {\matrix{ n \cr 0 \cr } } \right)\left( {\matrix{ n \cr k \cr } } \right) - {2^{^{k - 1}\left( {\matrix{ n \cr 2 \cr } } \right)}}\left( {\matrix{ n \cr 1 \cr } } \right)\left( {\matrix{ {n - 1} \cr {k - 1} \cr } } \right)$
$ + {2^{k - 2}}\left( {\matrix{ {n - 2} \cr {k - 2} \cr } } \right) - .....{\left( { - 1} \right)^k}\left( {\matrix{ n \cr k \cr } } \right)\left( {\matrix{ {n - k} \cr 0 \cr } } \right) = {\left( {\matrix{ n \cr k \cr } } \right)^ \cdot }$
2002 JEE Advanced Numerical
IIT-JEE 2002
Use mathematical induction to show that
${\left( {25} \right)^{n + 1}} - 24n + 5735$ is divisible by ${\left( {24} \right)^2}$ for all $ = n = 1,2,...$
2000 JEE Advanced Numerical
IIT-JEE 2000
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 heads occur. Prove that ${p_1} = 1,\,\,{p_2} = 1 - {p^2}$ and ${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$ for all $n \ge 3.$

Prove by induction on, that ${p_n} = A{\alpha ^n} + B{\beta ^n}$ for all $n \ge 1,$ where $\alpha $ and $\beta $ are the roots of quadratic equation ${x^2} - \left( {1 - p} \right)x - p\left( {1 - p} \right) = 0$ and $A = {{{p^2} + \beta - 1} \over {\alpha \beta - {\alpha ^2}}},B = {{{p^2} + \alpha - 1} \over {\alpha \beta - {\beta ^2}}}.$

2000 JEE Advanced Numerical
IIT-JEE 2000
For any positive integer $m$, $n$ (with $n \ge m$), let $\left( {\matrix{ n \cr m \cr } } \right) = {}^n{C_m}$
Prove that $\left( {\matrix{ n \cr m \cr } } \right) + \left( {\matrix{ {n - 1} \cr m \cr } } \right) + \left( {\matrix{ {n - 2} \cr m \cr } } \right) + ........ + \left( {\matrix{ m \cr m \cr } } \right) = \left( {\matrix{ {n + 1} \cr {m + 2} \cr } } \right)$

Hence or otherwise, prove that $\left( {\matrix{ n \cr m \cr } } \right) + 2\left( {\matrix{ {n - 1} \cr m \cr } } \right) + 3\left( {\matrix{ {n - 2} \cr m \cr } } \right) + ........ + \left( {n - m + 1} \right)\left( {\matrix{ m \cr m \cr } } \right) = \left( {\matrix{ {n + 2} \cr {m + 2} \cr } } \right).$.

2000 JEE Advanced Numerical
IIT-JEE 2000
Let $a,\,b,\,c$ be possitive real numbers such that ${b^2} - 4ac > 0$ and let ${\alpha _1} = c.$ Prove by induction that ${\alpha _{n + 1}} = {{a\alpha _n^2} \over {\left( {{b^2} - 2a\left( {{\alpha _1} + {\alpha _2} + ... + {\alpha _n}} \right)} \right)}}$ is well-defined and
${\alpha _{n + 1}} < {{{\alpha _n}} \over 2}$ for all $n = 1,2,....$ (Here, 'well-defined' means that the denominator in the expression for ${\alpha _{n + 1}}$ is not zero.)
2000 JEE Advanced Numerical
IIT-JEE 2000
For every possitive integer $n$, prove that
$\sqrt {\left( {4n + 1} \right)} < \sqrt n + \sqrt {n + 1} < \sqrt {4n + 2}.$
Hence or otherwise, prove that $\left[ {\sqrt n + \sqrt {\left( {n + 1} \right)} } \right] = \left[ {\sqrt {4n + 1} \,\,} \right],$
where $\left[ x \right]$ denotes the gratest integer not exceeding $x$.
1999 JEE Advanced Numerical
IIT-JEE 1999
Let $n$ be any positive integer. Prove that $$\sum\limits_{k = 0}^m {{{\left( {\matrix{ {2n - k} \cr k \cr } } \right)} \over {\left( {\matrix{ {2n - k} \cr n \cr } } \right)}}.{{\left( {2n - 4k + 1} \right)} \over {\left( {2n - 2k + 1} \right)}}{2^{n - 2k}} = {{\left( {\matrix{ n \cr m \cr } } \right)} \over {\left( {\matrix{ {2n - 2m} \cr {n - m} \cr } } \right)}}{2^{n - 2m}}} $$

for each non-be gatuve integer $m \le n.$ $\,\left( {Here\left( {\matrix{ p \cr q \cr } } \right) = {}^p{C_q}} \right).$

1998 JEE Advanced Numerical
IIT-JEE 1998
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 $p$ does not divide $r$, $p$ divides ${}^{np}{C_r},$

[Hint: You may use the fact that ${\left( {1 + x} \right)^{\left( {m + 1} \right)p}} = {\left( {1 + x} \right)^p}{\left( {1 + x} \right)^{mp}}$]

1997 JEE Advanced Numerical
IIT-JEE 1997
Let $0 < {A_i} < n$ for $i = 1,\,2....,\,n.$ Use mathematical induction to prove that $$\sin {A_1} + \sin {A_2}....... + \sin {A_n} \le n\,\sin \,\,\left( {{{{A_1} + {A_2} + ...... + {A_n}} \over n}} \right)$$

where $ \ge 1$ is a natural number. {You may use the fact that $p\sin x + \left( {1 - p} \right)\sin y \le \sin \left[ {px + \left( {1 - p} \right)y} \right],$ where $0 \le p \le 1$ and $0 \le x,y \le \pi .$}

1996 JEE Advanced Numerical
IIT-JEE 1996
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 ${2^{n + 3}}$.
1994 JEE Advanced Numerical
IIT-JEE 1994
If $x$ is not an integral multiple of $2\pi $ use mathematical induction to prove that : $$\cos x + \cos 2x + .......... + \cos nx = \cos {{n + 1} \over 2}x\sin {{nx} \over 2}\cos ec{x \over 2}$$
1994 JEE Advanced Numerical
IIT-JEE 1994
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 $a_0^2 - a_1^2 + a_2^2...... + {a_{2n}}{}^2 = {a_n}$
1993 JEE Advanced Numerical
IIT-JEE 1993
Using mathematical induction, prove that
${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1/\left( {{n^2} + n + 1} \right)} \right\} = {\tan ^{ - 1}}\left\{ {n/\left( {n + 2} \right)} \right\}$
1993 JEE Advanced Numerical
IIT-JEE 1993
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 an even positive integer.
1992 JEE Advanced Numerical
IIT-JEE 1992
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 ${a_k} = 1$ for all $k \ge n,$ then show that ${b_n} = {}^{2n + 1}{C_{n + 1}}$
1992 JEE Advanced Numerical
IIT-JEE 1992
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 show that ${\alpha ^n} + {\beta ^n}.$
(i) is an integer and (ii) is not divisible by $p$
1991 JEE Advanced Numerical
IIT-JEE 1991
Using induction or otherwise, prove that for any non-negative integers $m$, $n$, $r$ and $k$ ,
$\sum\limits_{m = 0}^k {\left( {n - m} \right)} {{\left( {r + m} \right)!} \over {m!}} = {{\left( {r + k + 1} \right)!} \over {k!}}\left[ {{n \over {r + 1}} - {k \over {r + 2}}} \right]$
1990 JEE Advanced Numerical
IIT-JEE 1990
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 $n$
1989 JEE Advanced Numerical
IIT-JEE 1989
Prove that
${C_0} - {2^2}{C_1} + {3^2}{C_2}\,\, - \,..... + {\left( { - 1} \right)^n}{\left( {n + 1} \right)^2}{C_n} = 0,\,\,\,\,n > 2,\,\,$ where ${C_r} = {}^n{C_r}.$
1989 JEE Advanced Numerical
IIT-JEE 1989
Using mathematical induction, prove that ${}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}{C_k},$
where $m,\,n,\,k$ are positive integers, and ${}^p{C_q} = 0$ for $p < q.$
1988 JEE Advanced Numerical
IIT-JEE 1988
Let $R$ $ = {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}$ and $f = R - \left[ R \right],$ where [ ] denotes the greatest integer function. Prove that $Rf = {4^{2n + 4}}$
1987 JEE Advanced Numerical
IIT-JEE 1987
Prove by mathematical induction that $ - 5 - {{\left( {2n} \right)!} \over {{2^{2n}}{{\left( {n!} \right)}^2}}} \le {1 \over {{{\left( {3n + 1} \right)}^{1/2}}}}$ for all positive integers $n$.
1985 JEE Advanced Numerical
IIT-JEE 1985
Use method of mathematical induction ${2.7^n} + {3.5^n} - 5$ is divisible by $24$ for all $n > 0$
1984 JEE Advanced Numerical
IIT-JEE 1984
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 integer $n$.
1984 JEE Advanced Numerical
IIT-JEE 1984
Given ${s_n} = 1 + q + {q^2} + ...... + {q^2};$
${S_n} = 1 + {{q + 1} \over 2} + {\left( {{{q + 1} \over 2}} \right)^2} + ........ + {\left( {{{q + 1} \over 2}} \right)^n}\,\,\,,q \ne 1$
Prove that ${}^{n + 1}{C_1} + {}^{n + 1}{C_2}{s_1} + {}^{n + 1}{C_3}{s_2} + ..... + {}^{n + 1}{C_n}{s_n} = {2^n}{S_n}$
1983 JEE Advanced Numerical
IIT-JEE 1983
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 two at a time, represented $\sum\limits_{0 \le i < j \le n} {\sum {{C_i}{C_j}} } $ is equal to ${2^{2n - 1}} - {{\left( {2n} \right)!} \over {2{{\left( {n!} \right)}^2}}}$
1983 JEE Advanced Numerical
IIT-JEE 1983
Use mathematical Induction to prove : If $n$ is any odd positive integer, then $n\left( {{n^2} - 1} \right)$ is divisible by 24.
1982 JEE Advanced Numerical
IIT-JEE 1982
Prove that ${7^{2n}} + \left( {{2^{3n - 3}}} \right)\left( {3n - 1} \right)$ is divisible by 25 for any natural number $n$.
1979 JEE Advanced Numerical
IIT-JEE 1979
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 ${C_r} = {{\left( {2n} \right)\,!} \over {r!\left( {2n - r} \right)!}}\,\,\,\,\,r = 0,1,2,\,............,2n$
Prove that ${C_1}^2 - 2{C_2}^2 + 3{C_3}^2 - ............ - 2n{C_{2n}}^2 = {\left( { - 1} \right)^n}n{C_n}.$
1997 JEE Advanced Numerical
IIT-JEE 1997
The sum of the rational terms in the expansion of ${\left( {\sqrt 2 + {3^{1/5}}} \right)^{10}}$ is ...............
1994 JEE Advanced Numerical
IIT-JEE 1994
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 value of $n$ is ................
1983 JEE Advanced Numerical
IIT-JEE 1983
If ${\left( {1 + ax} \right)^n} = 1 + 8x + 24{x^2} + .....$ then $a=..........$ and $n =............$
1982 JEE Advanced Numerical
IIT-JEE 1982
The sum of the coefficients of the plynomial ${\left( {1 + x - 3{x^2}} \right)^{2163}}$ is ...............
1982 JEE Advanced Numerical
IIT-JEE 1982
The larger of ${99^{50}} + {100^{50}}$ and ${101^{50}}$ is ..............