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Question
If \[T_2 / T_3\] in the expansion of \[\left( a + b \right)^n \text{ and } T_3 / T_4\] in the expansion of \[\left( a + b \right)^{n + 3}\] are equal, then n =
Options
3
4
5
6
Solution
5
\[\text{ In the expansion} (a + b )^n , \text{ we have } \]
\[\frac{T_2}{T_3} = \frac{^{n}{}{C}_1 a^{n - 1} \times b^1}{^{n}{}{C}_2 a^{n - 2} \times b^2}\]
\[\text{ In the expansion } (a + b )^{n + 3} , \text{ we have } \]
\[\frac{T_3}{T_4} = \frac{^{n + 3}{}{C}_2 a^{n + 1} b^2}{^{n + 3}{}{C}_3 a^n b^3}\]
\[\text{ Thus, we have } \]
\[\frac{T_2}{T_3} = \frac{T_3}{T_4}\]
\[ \Rightarrow \frac{^{n}{}{C}_1 a}{^{n}{}{C}_2 b} = \frac{^{n + 3}{}{C}_2 a}{^{n + 3}{}{C}_3 b}\]
\[ \Rightarrow \frac{2}{n - 1} = \frac{3}{n + 1}\]
\[ \Rightarrow 2n + 2 = 3n - 3\]
\[ \Rightarrow n = 5\]
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