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(B) The coefficients of the $2^{	ext{nd}}$,$3^{	ext{rd}}$,and $4^{	ext{th}}$ terms in the expansion of $(1+x)^n$ are given by $\binom{n}{1}$,$\bino

Vedclass · Accepted Answer

$7$

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(B) The coefficients of the $2^{	ext{nd}}$,$3^{	ext{rd}}$,and $4^{	ext{th}}$ terms in the expansion of $(1+x)^n$ are given by $\binom{n}{1}$,$\binom{n}{2}$,and $\binom{n}{3}$ respectively.Since these are in $A$.$P$.,we have $2 \binom{n}{2} = \binom{n}{1} + \binom{n}{3}$.Expanding the binomial coefficients: $2 	imes \frac{n(n-1)}{2} = n + \frac{n(n-1)(n-2)}{6}$.Dividing by $n$ (since $n > 0$): $n-1 = 1 + \frac{(n-1)(n-2)}{6}$.Multiplying by $6$: $6n - 6 = 6 + (n^2 - 3n + 2)$.Rearranging the terms: $n^2 - 9n + 14 = 0$.Factoring the quadratic equation: $(n-7)(n-2) = 0$.Thus,$n = 7$ or $n = 2$.For the $4^{	ext{th}}$ term to exist,$n$ must be at least $3$,so $n = 7$.

Let $n$ be a positive integer. If the coefficients of $2^{\text{nd}}$,$3^{\text{rd}}$,and $4^{\text{th}}$ terms in the expansion of $(1+x)^n$ are in $A$.$P$.,then the value of $n$ is:

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