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  4. 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 :-

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Q. 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 :-

JEE MainJEE Main 2020Binomial Theorem

Solution:

Let $n+5=N$
$N_{C_{t-1}}: N_{C_{t}}: N_{C_{n-1}}=5: 10: 14$
$\Rightarrow \frac{N_{C_{s}}}{N_{C_{t-1}}}=\frac{N+1-r}{r}=2$
$ \frac{N_{C_{t+1}}}{N_{C_{t}}}=\frac{N-r}{r+1}=\frac{7}{5}$
$\Rightarrow r=4, N=11$
$\Rightarrow (1+x)^{11}$
Largest coefficient $={ }^{11} C _{6}=462$