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Sum of N Terms of an AP

The expansion...

Question

The expansion of ${(1 + x)}^{n}$ has 3 consecutive terms with coefficient in the ratio $1 : 2 : 3$ and can be written in the form $^{n} C_{k};^{n} C_{k + 1} :^{n} C_{k + 2} .$ . The sum of all possible values of $n + k$ is

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Solution

The correct option is A 18
Given
$^{n} C_{k} :^{n} C_{k + 1} :^{n} C_{k + 2} = 1 : 2 : 3$
$nCknCk+1=12⇒n!(n−k−1)(n−k)!x(k)!×(k+1)!n!=dfrac12$
$\Rightarrow \frac{(k + 1)}{(n - k)} = \frac{1}{2}$
$\Rightarrow 2 k + 2 = n - k$
$\Rightarrow n = 3 k + 2$ ------- $(1)$
$\frac{^{n} C_{k + 1}}{^{n} C_{k + 2}} = \frac{2}{3} \Rightarrow \frac{n! (n - k - 2)!}{(n - k - 1)! (k + 1)!} \Rightarrow \frac{2}{3}$
$\Rightarrow \frac{(k + 2)}{(n - k - 1)} = \frac{2}{3}$
$\Rightarrow 3 k + 6 = 2 n - 2 k - 2$
$\Rightarrow 5 k + 8 = 2 n$
On solving $6 k + 9 = 5 k + 8 \Rightarrow k = 4 \Rightarrow n = x 4 + 2$
$∴ n + k = 14 + 4 = 18$
$n = 14$

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