Question : If $1^{2} + 2^{2} + 3^{2} + . . . . . . + p^{2} = \frac{p (p + 1) (2 p + 1)}{6}$ , then $1^{2} + 3^{2} + 5^{2} + . . . . . . + 17^{2}$ is equal to:
Option 1: 1785
Option 2: 1700
Option 3: 980
Option 4: 969

Question

Question : If $\cos^{4} α - \sin^{4} α = \frac{5}{6}$ , then the value of $2 \cos^{2} α - 1$ is:

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Answer 1

Correct Answer: 969

Solution : Given: $1^{2} + 2^{2} + 3^{2} + . . . . . . + p^{2} = \frac{p (p + 1) (2 p + 1)}{6}$
So, $1^{2} + 2^{2} + 3^{2} + . . . . . . + 17^{2} = \frac{17 (17 + 1) (2 \times 17 + 1)}{6}$ -----------------(1)
Also, $2^{2} + 4^{2} + 6^{2} + . . . . . . + 16^{2} = 2^{2} (1^{2} + 2^{2} + 3^{2} + . . . . . . + 8^{2})$
$\Rightarrow 2^{2} + 4^{2} + 6^{2} + . . . . . . + 16^{2} = 2^{2} (\frac{8 (8 + 1) (2 \times 8 + 1)}{6})$ ----------------(2)
Subtracting equation (2) from equation (1), we get,
$1^{2} + 3^{2} + 5^{2} + . . . . . . + 17^{2} = \frac{17 (17 + 1) (2 \times 17 + 1)}{6} - 2^{2} (\frac{8 (8 + 1) (2 \times 8 + 1)}{6})$
$∴ 1^{2} + 3^{2} + 5^{2} + . . . . . . + 17^{2} = 969$
Hence, the correct answer is 969.

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Question : If 12+22+32+......+ p2=p(p+1)(2p+1)6, then 12+32+52+......+172 is equal to:Option 1: 1785Option 2: 1700Option 3: 980Option 4: 969

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Question : If $1^{2} + 2^{2} + 3^{2} + . . . . . . + p^{2} = \frac{p (p + 1) (2 p + 1)}{6}$ , then $1^{2} + 3^{2} + 5^{2} + . . . . . . + 17^{2}$ is equal to:
Option 1: 1785
Option 2: 1700
Option 3: 980
Option 4: 969