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Question
The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find n.
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Solution
\[\text{ Coefficients of the 5th, 6th and 7th terms in the given expansion are } ^{n}{}{C}_4 ,^{n}{}{C}_5 \text{ and } ^{n}{}{C}_6 \]
\[\text{ These coefficients are in AP } . \]
\[\text{ Thus, we have} \]
\[2 ^{n}{}{C}_5 = ^{n}{}{C}_4 +^{n}{}{C}_6 \]
\[\text{ On dividing both sides by }^{n}{}{C}_5 , \text{ we get: } \]
\[2 = \frac{^{n}{}{C}_4}{^{n}{}{C}_5} + \frac{^{n}{}{C}_6}{^{n}{}{C}_5}\]
\[ \Rightarrow 2 = \frac{5}{n - 4} + \frac{n - 5}{6}\]
\[ \Rightarrow 12n - 48 = 30 + n^2 - 4n - 5n + 20\]
\[ \Rightarrow n^2 - 21n + 98 = 0\]
\[ \Rightarrow (n - 14)(n - 7) = 0\]
\[ \Rightarrow n = 7 or 14\]
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