If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$, then the largest coefficient in this expansion is :
Correct Option: 1
Consider the three consecutive coefficients of
$(1+x)^{n+5}$ be ${ }^{n+5} C_{r},{ }^{n+5} C_{r+1},{ }^{n+5} C_{r+2}$
\because \frac{{ }^{n+5} C_{r}}{{ }^{n+5} C_{r+1}}=\frac{1}{2} (Given)
$\Rightarrow \frac{r+1}{n+5-r}=\frac{1}{2} \Rightarrow 3 r=n+3$ ..........(1)
and $\frac{{ }^{n+5} C_{r+1}}{{ }^{n+5} C_{r+2}}=\frac{5}{7}$
$\Rightarrow \frac{r+2}{n+4-r}=\frac{5}{7} \Rightarrow 12 r=5 n+6$ .........(2)
Solving (1) and (2) we get $r=4$ and $n=6$
$\therefore$ Largest coefficient in the expansion is ${ }^{11} C_{6}=462$.
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