If the coefficients of three consecutive terms in the expansion of $(1 + x)^{n}$ are $165, 330$ and $462$ respectively, then the value of $n$ is

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Solution

Let the consecutive terms be $(r + 1)^{t h}, (r + 2)^{t h}, (r + 3)^{t h}$ in expansion of $(1 + x)^{n}$
Then,
$^{n} C_{r} = 165^{n} C_{r + 1} = 330^{n} C_{r + 2} = 462$
Now,
$\frac{^{n} C_{r + 1}}{^{n} C_{r}} = \frac{330}{165} \Rightarrow \frac{n - r}{r + 1} = 2 \Rightarrow n - r = 2 r + 2 \Rightarrow n = 3 r + 2 \dots (1)$
Also,
$\frac{^{n} C_{r + 2}}{^{n} C_{r + 1}} = \frac{462}{330} \Rightarrow \frac{n - r - 1}{r + 2} = \frac{231}{165}$
From equation $(1)$ , we get
$\Rightarrow \frac{2 r + 1}{r + 2} = \frac{231}{165} \Rightarrow 330 r + 165 = 231 r + 462 ∴ r = 3 \Rightarrow n = 11$

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