Answer 1

 7

Coefficients of the 2nd, 3rd  and 4th terms in the given expansion are:

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