If coefficients of {(2r + 1)^{th}} term and {(r + 2)^{th}} term are equal in expansion of {(1 + x)^{

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7.Binomial Theorem

easy

If coefficients of ${(2r + 1)^{th}}$ term and ${(r + 2)^{th}}$ term are equal in the expansion of ${(1 + x)^{43}},$ then the value of $r$ will be

$14$

$15$

$13$

$16$

Solution

(a) Coefficient of ${(2r + 1)^{th}}$ term in expansion of $(1 + x)^{43}$ = ${}^{43}{C_{2r}}$ and coefficient of ${(r + 2)^{th}}$ term = coefficient of ${\{ (r + 1) + 1\} ^{th}}$ term = ${}^{43}{C_{r + 1}}$

According to question ${}^{43}{C_{2r}} = {}^{43}{C_{r + 1}}$

$ = {}^{43}{C_{43 – (r + 1)}}$ then $2r = 43 – (r + 1)$or $3r = 42$ or $r = 14$.

Standard 11

Mathematics

If ${x^m}$occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is

medium

If the second term of the expansion ${\left[ {{a^{\frac{1}{{13}}}}\,\, + \,\,\frac{a}{{\sqrt {{a^{ – 1}}} }}} \right]^n}$ is $14a^{5/2}$ then the value of $\frac{{^n{C_3}}}{{^n{C_2}}}$ is :

normal

The term independent of $x$ in ${\left( {\sqrt x – \frac{2}{x}} \right)^{18}}$ is

easy

If the coefficients of the three consecutive terms in the expansion of $(1+ x )^{ n }$ are in the ratio $1: 5: 20$, then the coefficient of the fourth term is $…………$.

hard

(JEE MAIN-2023)

In the expansion of the following expression $1 + (1 + x) + {(1 + x)^2} + ….. + {(1 + x)^n}$ the coefficient of ${x^k}(0 \le k \le n)$ is