If the coefficients of the $2^{n d}, 3^{r d}$ and $4^{t h}$ terms in the expansion of $(1 + x)^{n}$ , $n ϵ N$ ,are in $A P$ then $n$ is

7

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14

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2

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none of these

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Solution

The correct option is A $7$
Hence $^{n} C_{1}$ , $^{n} C_{2}$ , $^{n} C_{3}$ are in A.P
For $^{n} C_{r - 1}$ , $^{n} C_{r}$ , $^{n} C_{r + 1}$ to be in A.P it must follow that
$(n - 2 r)^{2} = n + 2$
Comparing with the above question, we get
$r = 2$
Hence $(n - 4)^{2} = n + 2$
$\Rightarrow n^{2} - 8 n + 16 = n + 2$
$\Rightarrow n^{2} - 9 n + 14 = 0$
$\Rightarrow (n - 7) (n - 2) = 0$
$n = 7$ as $n = 2$ is not possible.

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