The sum to 20 terms of the series 2 · 2 2 - 3 2 + 2 · 4 2 - 5 2 + 2 · 6 2 - . . . . . . . . . . is equal to _______ . [2023]

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Solution

Q.
The sum to 20 terms of the series $2 \cdot 2^{2} - 3^{2} + 2 \cdot 4^{2} - 5^{2} + 2 \cdot 6^{2} - . . . . . . . . . .$ is equal to _______ . [2023]

Ans.
(1310)

$Let S = 2 \cdot 2^{2} - 3^{2} + 2 \cdot 4^{2} - 5^{2} + 2 \cdot 6^{2} + \dots$

$= (2 \cdot 2^{2} + 2 \cdot 4^{2} + 2 \cdot 6^{2} + \dots 10 terms)$ $- (3^{2} + 5^{2} + 7^{2} + \dots + 10 terms)$

$= 8 (1^{2} + 2^{2} + 3^{2} + \dots + 10 terms)$ $- (1^{2} + 3^{2} + 5^{2} + 7^{2} + \dots + 11 terms) + 1$

$= 8 [\frac{10 \times 11 \times 21}{6}] - \frac{11}{3} [23 \times 21] + 1$

$= 3080 - 1771 + 1 = 1310$

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