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Question
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 n.
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Solution
Coefficients of the 2nd, 3rd and 4th terms in the given expansion are:
\[^{n}{}{C}_1 , ^{n}{}{C}_2 \text{ and } ^{n}{}{C}_3 \]
\[\text{ We have: } \]
\[2 \times ^{n}{}{C}_2 = ^{n}{}{C}_1 + ^{n}{}{C}_3 \]
\[\text{ Dividing both sides by} ^{n}{}{C}_2 , \text{ we get: } \]
\[2 = \frac{^{n}{}{C}_1}{^{n}{}{C}_2} + \frac{^{n}{}{C}_3}{^{n}{}{C}_2}\]
\[ \Rightarrow 2 = \frac{2}{n - 1} + \frac{n - 2}{3}\]
\[ \Rightarrow 6n - 6 = 6 + n^2 + 2 - 3n\]
\[ \Rightarrow n^2 - 9n + 14 = 0\]
\[ \Rightarrow n = 7 \left( \because n \neq 2 \text{ as } 2 > 3 \text{ in the 4th term } \right)\]
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