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Three consecutive coefficients in the expansion of (1 + x)ⁿ are 165, 330 and 462…

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Three consecutive coefficients in the expansion of $(1 + x)^n$ are $165$ , $330$ and $462$ . Find $n$ , and find which term of the expansion the middle coefficient belongs to.

Need a reminder? The key ideaShow

For binomial coefficients, ${}^{n}C_{r} = {}^{n}C_{n - r}$ , and if ${}^{n}C_{a} = {}^{n}C_{b}$ with $a \ne b$ then $a + b = n$ . Pascal's rule is ${}^{n}C_{r - 1} + {}^{n}C_{r} = {}^{n + 1}C_{r}$ . Neighbours satisfy $\dfrac{{}^{n}C_{r}}{{}^{n}C_{r - 1}} = \dfrac{n - r + 1}{r}$ , and $r \cdot {}^{n}C_{r} = n \cdot {}^{n - 1}C_{r - 1}$ .

The worked solution

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Three consecutive coefficients in the expansion of (1 + x)ⁿ are 165, 330 and 462…

The worked solution

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