If for some positive integer n , the coefficients of three consecutive terms in the binomial expansion of 1…

Let three consecutive terms are $T_r, T_{r+1}, T_{r+2}$
So, $\frac{{}^{n+5}C_{r-1}}{{}^{n+5}C_r}=\frac{5}{10}$ and $\frac{{}^{n+5}C_r}{{}^{n+5}C_{r+1}}=\frac{10}{14}$

Using property $\frac{{}^{n}C_r}{{}^{n}C_{r-1}}=\frac{n-r+1}{r}$, we have

$\frac{\left(n+5\right)-r+1}{r}=2$ and $\frac{\left(n+5\right)-\left(r+1\right)+1}{r+1}=\frac{7}{5}$

$\Rightarrow n-3r+6=0$ and $5n-12r+18=0$

Solving both equations, $n=6, r=4$
Hence, the greatest coefficient will be of middle term, which is $={}^{n+5}C_5={}^{11}C_5=462$