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 $\frac{r+2}{n+5-r-1}=\frac{5}{7}$

$\Rightarrow 3r=n+3$  and $12r=5n+6$

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