If the coefficients of $2^{n d}, 3^{r d}$ and $4^{t h}$ terms of the expansion of $(1 + x)^{2 n}$ are in $A . P$ , then the value of $2 n^{2} - 9 n + 7$ is

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Solution

The correct option is B 0
Let $^{2 n} C_{r - 1},^{2 n} C_{r},^{2 n} C_{r + 1}$ be in AP
Hence $T_{r}, T_{r + 1}, T_{r + 2}$ be in AP
Therefore condition for the following terms to be in A.P is
$(n - 2 r)^{2} = n + 2$ ...(i)
In the above question
$T_{r} = T_{2}$
Hence $r = 2$
Substituting in (i), we get
$(2 n - 4)^{2} = 2 n + 2$
$4 n^{2} - 16 n + 16 = 2 n + 2$
$4 n^{2} - 18 n + 14 = 0$
$2 n^{2} - 9 n + 7 = 0$ ...(Answer)
$(2 n - 7) (n - 1) = 0$
Hence answer is option A

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