For some , let the coefficients of the and terms in the binomial expansion of be in A.P. Then the largest coefficient in the expansion of is: [2025]

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Solution

Q.
For some $n \neq 10$ , let the coefficients of the $5^{th}, 6^{th}$ and $7^{th}$ terms in the binomial expansion of ${(1 + x)}^{n + 4}$ be in A.P. Then the largest coefficient in the expansion of ${(1 + x)}^{n + 4}$ is: [2025]

1 35

2 20

3 10

4 70

Ans.
(1)

Coefficient of $5^{th}, 6^{th}$ and $7^{th}$ terms of the expansion of ${(1 + x)}^{n + 4}$ are ${}^{n + 4}C_{4}, {}^{n + 4}C_{5}$ and ${}^{n + 4}C_{6}$ respectively. As ${}^{n + 4}C_{4}, {}^{n + 4}C_{5}$ and ${}^{n + 4}C_{6}$ are in A.P.

$\Rightarrow 2 \times {}^{n + 4}C_{5} = {}^{n + 4}C_{4} + {}^{n + 4}C_{6}$

$\Rightarrow \frac{2}{5 (n - 1)} = \frac{1}{n (n - 1)} + \frac{1}{30} \Rightarrow \frac{2}{5 (n - 1)} = \frac{30 + n^{2} - n}{30 n (n - 1)}$

$\Rightarrow 12 n = 30 + n^{2} - n \Rightarrow n^{2} - 13 n + 30 = 0$

$\Rightarrow (n - 3) (n - 10) = 0 \Rightarrow n = 3$ [ $∵ n \neq 10$ ]

Here, n + 4 = 3 + 4 = 7

$∴$ Largest binomial coefficient in expansion ${(1 + x)}^{7}$ = Coefficient of middle term = ${}^{7}C_{4} or {}^{7}C_{3} = 35$ .

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