If the coefficients of 2nd, 3rd and 4th terms in the expansion of $(1 + x)^{n}$ are in A.P., then find the value of

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Solution

Coefficients of the 2nd, 3rd and 4th term in the given expansion are:

$^{n} C_{1},^{n} C_{2}$ and $^{n} C_{3}$

We have:

$2 \times^{n} C_{2} =^{n} C_{1} +^{n} C_{3}$

Dividing both sides by $^{n} C_{2}$ , we get:

$2 = \frac{^{n} C_{1}}{^{n} C_{2}} + \frac{^{n} C_{3}}{^{n} C_{2}}$

$\Rightarrow 2 = \frac{2}{n - 1} + \frac{n - 2}{3}$

$\Rightarrow 6 n - 6 = 6 + n^{2} + 2 - 3 n$

$\Rightarrow n^{2} - 9 n + 14 = 0$

$\Rightarrow n = 7 [∵ n \neq 2$ as 2 > 3 in the 4th term]

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