If $a_{n}$ denotes the coefficient of $x^{n}$ in $P (x) = (1 + x + 2 x^{2} + . . . . + 25 x^{25})^{2}$ , find $\frac{a_{5}}{5}$ .

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Solution

$P (x) = (1 + x + 2 x^{2} + 3 x^{3} + . . . . + 25 x^{25}) (1 + x + 2 x^{2} + 3 x^{3} + . . . . + 25 x^{25})$
Looking at the terms that can have $x^{5}$ , we get
$a_{5} = coeff of x^{5} = 1.5 + 1.4 + 2.3 + 3.2 + 4.1 + 5 = 30$
$\Rightarrow \frac{a_{5}}{5} = 6$

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