For a positive integer n, ( 1 + 1 / x )^n is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12, then n is equal to________.

Let, three consecutive coefficients are ^nC_r - 1,^nC_r,^nC_r + 1 ^nC_r - 1:^nC_r:^nC_r + 1 = 2:5:12 Now, ^nC_r - 1 / ^nC_r = 2 / 5 => 7r = 2n + 2 ...(i) ^nC_r / ^nC_r + 1 = 5 / 12 => 7r = 5n - 12 ...(ii) On solving (i) and (ii) we get n = 118

For a positive integer n 1 1 x n is expanded in increas... | null (Mathematics)

Let, three consecutive coefficients are

{}^n{C_{r - 1}},{}^n{C_r},{}^n{C_{r + 1}}

{}^n{C_{r - 1}}:{}^n{C_r}:{}^n{C_{r + 1}} = 2:5:12

Now,

{{{}^n{C_{r - 1}}} \over {{}^n{C_r}}} = {2 \over 5}

\Rightarrow 7r = 2n + 2 \quad ...(i)

{{{}^n{C_r}} \over {{}^n{C_{r + 1}}}} = {5 \over {12}}

\Rightarrow 7r = 5n - 12 \quad ...(ii)

On solving (i) and (ii) we get n = 118

Explanation

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