If the coefficients of three consecutive terms of (1+x)n+5 are in the ratio 5:10:14, then n is equal to

Let three consecutive coefficients of $(1 + x)^{n + 5}$ be $^{n + 5} C_{r},^{n + 5} C_{r + 1},^{n + 5} C_{r + 2}$

We are given

$\frac{^{n + 5} C_{r}}{^{n + 5} C_{r + 1}} = \frac{5}{10}$ and $\frac{^{n + 5} C_{r + 1}}{^{n + 5} C_{r + 2}} = \frac{10}{14}$

$\Rightarrow \frac{r + 1}{n + 5 - r} = \frac{1}{2}$ and $\frac{r + 2}{n + 5 - r - 1} = \frac{5}{7}$

$\Rightarrow 3 r = n + 3$ and $12 r = 5 n + 6$

$∴ 4 (n + 3) = 5 n + 6 \Rightarrow n = 6$

If the coefficients of three consecutive terms of $(1 + x)^{n + 5}$ are in the ratio $5 : 10 : 14$ , then n is equal to