upto 40 terms is equal to [2025]

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Solution

Q.
$1 + 3 + 5^{2} + 7 + 9^{2} + . . .$ upto 40 terms is equal to [2025]

1 41880

2 33980

3 40870

4 43890

Ans.
(1)

$1 + 3 + 5^{2} + 7 + 9^{2} + . . . 40$ terms

$= (1^{2} + 5^{2} + 9^{2} + . . . 20 terms) + (3 + 7 + 11 + . . . 20 terms)$

$= \sum_{r = 1}^{20} {(4 r - 3)}^{2} + \sum_{r = 1}^{20} (4 r - 1)$

$= \sum_{r = 1}^{20} (16 r^{2} - 20 r + 8)$

$= 16 \sum_{r = 1}^{20} r^{2} - 20 \sum_{r = 1}^{20} r + 8 \sum_{r = 1}^{20} 1$

$= \frac{16 \times 20 \times 21 \times 41}{6} - \frac{20 \times 20 \times 21}{2} + 8 \times 20 = 41880$ .

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