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Arithmetic Progression

If the coeffi...

Question

If the coefficients of $x^{7}$ and $x^{8}$ in are equal, then $n$ is equal to

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Solution

It is given, coefficient of $x^{7} =$ Coefficient of $x^{8}$
$^{n} C_{7}^{2 n - 7} {(\frac{1}{3})}^{7} =^{n} C_{8} 2^{n - 8} {(\frac{1}{3})}^{8}$
$\Rightarrow 6 (^{n} C_{7}) =^{n} C_{8}$
$\Rightarrow 48 = n - 7$
$\Rightarrow n = 55$

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