Let $n$ be a positive integer. If the coefficients of $2 n d, 3 r d$ and $4 t h$ terms in the expansion of ${(1 + x)}^{n}$ are in A.P., then the value of $n$ is .......

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Solution

Let, the coefficients of $2 n d, 3 r d$ and $4 t h$ terms in the
expansion of ${(1 + x)}^{n}$ is $^{n} C_{1},^{n} {C_{2},}^{n} C_{3} .$
According to given condition
$2 (^{n} C_{2}) =^{n} C_{1} +^{n} C_{3}$
$\Rightarrow 2 \frac{n (n - 1)}{1.2} = n + \frac{n (n - 1) (n - 2)}{1.2.3}$
$\Rightarrow n - 1 = 1 + \frac{(n - 1) (n - 2)}{6}$
$\Rightarrow n - 1 = 1 + \frac{n^{2} - 3 n + 2}{6}$
$\Rightarrow 6 n - 6 = 6 + n^{2} - 3 n + 2 \Rightarrow n^{2} - 09 n + 14 = 0$
$\Rightarrow (n - 2) (n - 7) = 0 \Rightarrow n = 2$ or $n = 7$
But $^{n} C_{3}$ is true for $n \geq 3,$ therefore $n = 7$ is the answer

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