Mathematical Induction and Binomial Theorem

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 val...

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...

For non-negative integers s and r, let\(\left( {\matrix{ s \cr r \cr } } \right) = \left\{ {\matrix{ {{{s!} \over {r!(s - r)!}}} & {if\,r \le \,s,} \cr 0 & {if\,r\, > \,s} \cr } } \right.\)For positive integers m and...

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 ..........

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}}\,\,$...

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

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

For \(r = 0,\,1,....,\) let \({A_r},\,{B_r}\) and \({C_r}\) denote, respectively, the coefficient of \({X^r}\) in the expansions of \({\left( {1 + x} \right)^{10}},\) \({\left( {1 + x} \right)^{20}}\) and \({\left( {1 + x} \right)^{30}}.\)...

The value of $$$\left( {\matrix{
{30} \cr
0 \cr

} } \right)\left( {\matrix{
{30} \cr
{10} \cr

} } \right) - \left( {\matrix{
{30} \cr
1 \cr

} } \right)\left( {\matrix{
{30} \cr
{11} \cr

} } \r...

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

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

} } \ri...

The sum $$\sum\limits_{i = 0}^m {\left( {\matrix{
{10} \cr
i \cr

} } \right)\left( {\matrix{
{20} \cr
{m - i} \cr

} } \right),\,\left( {where\left( {\matrix{
p \cr
q \cr

} } \right) = 0\,\,if\,\,p < q} \r...

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

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

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 ...

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 ...

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...

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} ...

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)} \ov...

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 \(-6\) respectively, then \(m\) is

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 ...

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

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 n...

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

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 \({2^{n + 3}}\).

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 ................

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 \(a_0^2 - a_1^2 + a_2^2...... + {a_{2n}}{}^2 = {a_n}\)

If \(x\) is not an integral multiple of \(2\pi\) use mathematical induction to prove that :
\($\cos x + \cos 2x + .......... + \cos nx = \cos {{n + 1} \over 2}x\sin {{nx} \over 2}\cos ec{x \over 2}\)$

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...

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.

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 \({b_n} = {}^{2n + 1}{C_{n + 1}}\)

The expansion \({\left( {x + {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5}\) is a polynomial of degree

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 no...

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...

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 \({}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}{C_k},\)
where \(m,\,n,\,k\) are positive integers, and \({}^p{C_q} = 0\) for \(p < q.\)

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 \({C_r} = {}^n{C_r}.\)

Let \(R\) \(= {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}\) and \(f = R - \left[ R \right],\) where [ ] denotes the greatest integer function. Prove that \(Rf = {4^{2n + 4}}\)

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 \(n\).

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}} \right){\mkern 1mu} !} \over {n!}}\left[ {C_0^2 - 2C_1^2 + 3C_2^2 - } \right......... + {\l...

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 integer \(n\).

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...

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 binomial expansion of \({\left( {1 + x} \right)^{2n}}\) are equal. Then

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 two at a time, represented \(\sum\limits_{0 \le i < j \le n} {\sum {{C_i}{C_j}} }\) is equ...

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

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