The value of $1^{2} + 3^{2} + 5^{2} +$ _____ $+ 25^{2}$ is:

1728

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1456

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2925

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4875

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Solution

The correct option is B $4875$
We have,
$1^{2} + 3^{2} + 5^{2} + . . . . .$ $+ 25^{2}$ $= (1^{2} + 2^{2} + 3^{2} + . . . . . + 25^{2}) - (2^{2} + 4^{2} + . . . . + 24^{2})$
or, $1^{2} + 3^{2} + 5^{2} + . . . . .$ $+ 25^{2}$ $= (1^{2} + 2^{2} + 3^{2} + . . . . . + 25^{2}) - 2^{2} (1^{2} + 2^{2} + . . . . + 12^{2})$
or, $1^{2} + 3^{2} + 5^{2} + . . . . .$ $+ 25^{2}$ $= \frac{25. (25 + 1) (2.25 + 1)}{6} - \frac{12. (12 + 1) (2.12 + 1)}{6} = 25.13.17 - 2.13.25 = 25.13.15 = 4875$

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