(D) The coefficients of the $2^{nd}$,$3^{rd}$,and $4^{th}$ terms in the expansion of $(1 + x)^n$ are $^nC_1$,$^nC_2$,and $^nC_3$ respectively.Since th

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(D) The coefficients of the $2^{nd}$,$3^{rd}$,and $4^{th}$ terms in the expansion of $(1 + x)^n$ are $^nC_1$,$^nC_2$,and $^nC_3$ respectively.Since these are in $A.P.$,we have $2(^nC_2) = ^nC_1 + ^nC_3$.Using the formula $^nC_r = \frac{n!}{r!(n-r)!}$,we get:$2 \times \frac{n(n-1)}{2} = n + \frac{n(n-1)(n-2)}{6}$.Dividing by $n$ (assuming $n \neq 0$):$(n-1) = 1 + \frac{(n-1)(n-2)}{6}$.$6(n-1) = 6 + (n^2 - 3n + 2)$.$6n - 6 = 6 + n^2 - 3n + 2$.$n^2 - 9n + 14 = 0$.Therefore,$n^2 - 9n = -14$.

Class 11 Mathematics Binomial Theorem General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

Medium

If coefficients of $2^{nd}$,$3^{rd}$ and $4^{th}$ terms in the binomial expansion of $(1 + x)^n$ are in $A.P.$,then $n^2 - 9n$ is equal to

A
$-7$
B
$7$
C
$14$
D
$-14$

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Binomial Theorem

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General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

The sum of the coefficients of the first three terms in the expansion of $(x - \frac{3}{x^2})^m$,where $x \neq 0$ and $m$ is a natural number,is $559$. Find the term of the expansion containing $x^3$. (in $x^3$)

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Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $(1+x-x^2)^6$ is

MediumAP EAMCET 2025

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If the coefficients of the $r^{th}$ term and the $(r + 4)^{th}$ term are equal in the expansion of $(1 + x)^{20}$,then the value of $r$ is:

Medium

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The coefficient of the term independent of $x$ in the expansion of $(1 + x + 2x^3)\left( \frac{3}{2}x^2 - \frac{1}{3x} \right)^9$ is

Difficult

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Let $\alpha > 0$ be the smallest number such that the expansion of $(x^{2/3} + 2x^{-3})^{30}$ has a term $\beta x^{-\alpha}$,where $\beta \in \mathbb{N}$. Then $\alpha$ is equal to $.............$.

DifficultJEE MAIN 2023

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If coefficients of $2^{nd}$,$3^{rd}$ and $4^{th}$ terms in the binomial expansion of $(1 + x)^n$ are in $A.P.$,then $n^2 - 9n$ is equal to

Similar Questions

The sum of the coefficients of the first three terms in the expansion of $(x - \frac{3}{x^2})^m$,where $x \neq 0$ and $m$ is a natural number,is $559$. Find the term of the expansion containing $x^3$. (in $x^3$)

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