$\sum_{i = 1}^{90} a i (i + 1)$ is equal to

\frac{a + 1}{4}

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\frac{a + 1}{2 \sqrt{2}}

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\frac{a - 1}{2 \sqrt{2}}

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None of the above

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Solution

The correct option is D None of the above
$\sum_{i = 1}^{90} a i (i + 1)$

$= \sum_{i = 1}^{90} a i^{2} + a i$

$= a \sum_{i = 1}^{90} i^{2} + a \sum_{i = 1}^{90} i$

$= {[a \frac{i (i + 1) (2 i + 1)}{6} + a \frac{i (i + 1)}{2}]}_{i = 90}$

$= a \frac{90.91.181}{6} + a \frac{90.91}{2}$

$= 251160 a$

Hence, the answer is $251160 a$ .

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