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

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Solution

The correct option is C 7
Here it is given that $T_{2}, T_{3}, T_{4}$ are in A.P.
Hence $^{n} C_{1},^{n} C_{2},^{n} C_{3}$ are in AP.
Let $^{n} C_{r - 1},^{n} C_{r},^{n} C_{r + 1}$ 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
$^{n} C_{r} =^{n} C_{2}$
Hence $r = 2$
Substituting in (i), we get
$(n - 4)^{2} = n + 2$
$n^{2} - 8 n + 16 = n + 2$
$n^{2} - 9 n + 14 = 0$
$(n - 7) (n - 2) = 0$
$n = 2$ and $n = 7$
Hence, answer is option C.

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