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Binomial Coefficients

Let n be a po...

Question

Let n be a positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of $(1 + x)^{n}$ are in AP, then the value of n is___

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Solution

The coefficients of 2nd, 3rd and 4th terms in the expansion of $(1 + x)^{n}$ is $^{n} C_{1},^{n} C_{2},^{n} C_{3}$
According to the given condition,
$2 (^{n} C_{2}) =^{n} C_{1} +^{n} C_{3} \Rightarrow 2 \frac{n (n - 1)}{1.2} = n + \frac{n (n - 1) (n - 2)}{1.2.3} \Rightarrow n - 1 = 1 + \frac{(n - 1) (n - 2)}{6} \Rightarrow n - 1 = 1 + \frac{n^{2} - 3 n + 2}{6} \Rightarrow 6 n - 6 = 6 + n^{2} - 3 n + 2 \Rightarrow n^{2} - 9 n + 14 = 0 \Rightarrow (n - 2) (n - 7) = 0 \Rightarrow n = 2, 7$

But $^{n} C_{3}$ is true for n $\geq$ 3, therefore n = 7 is the answer.

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