Answer 1

 5

\[\text{ In the expansion}  (a + b )^n , \text{ we have } \]

\[\frac{T_2}{T_3} = \frac{^{n}{}{C}_1 a^{n - 1} \times b^1}{^{n}{}{C}_2 a^{n - 2} \times b^2}\]

\[\text{ In the expansion } (a + b )^{n + 3} , \text{ we have } \]

\[\frac{T_3}{T_4} = \frac{^{n + 3}{}{C}_2 a^{n + 1} b^2}{^{n + 3}{}{C}_3 a^n b^3}\]

\[\text{ Thus, we have } \]

\[\frac{T_2}{T_3} = \frac{T_3}{T_4}\]

\[ \Rightarrow \frac{^{n}{}{C}_1 a}{^{n}{}{C}_2 b} = \frac{^{n + 3}{}{C}_2 a}{^{n + 3}{}{C}_3 b}\]

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

\[ \Rightarrow 2n + 2 = 3n - 3\]

\[ \Rightarrow n = 5\]