If 1-1-22-1-1-42-3-1-62-5-1-202-19- -220- then an ordered pair - - is equal to

1+(1 2^2·1)+(1 4^2·3)+(1 6^2·5)+…+(1 20^2·19) =11 (2^2·1+4^2·3+⋯20^2·19) =11 ∑_n=1^10(2n)^2(2n 1) =11 4∑_n=1^10(2n^3 n^2) =11 4(10^2(10+1)^22 10(10+1)(2×10+1)6) ...

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Byju's Answer

Standard XII

Mathematics

Logarithmic Differentiation

If 1+1-22·1+1...

Question

If $1 + (1 - 2^{2} \cdot 1) + (1 - 4^{2} \cdot 3) + (1 - 6^{2} \cdot 5) + \dots + (1 - 20^{2} \cdot 19) =$ $α - 220 β$ , then an ordered pair $(α, β)$ is equal to

(10, 97)

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(11, 103)

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(11, 97)

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(10, 103)

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Solution

The correct option is B $(11, 103)$
$1 + (1 - 2^{2} \cdot 1) + (1 - 4^{2} \cdot 3) + (1 - 6^{2} \cdot 5) + \dots + (1 - 20^{2} \cdot 19)$
$= 11 - (2^{2} \cdot 1 + 4^{2} \cdot 3 + \dots 20^{2} \cdot 19)$
$= 11 - 10 \sum n = 1 (2 n)^{2} (2 n - 1)$
$= 11 - 4 10 \sum n = 1 (2 n^{3} - n^{2})$
$= 11 - 4 (\frac{10^{2} (10 + 1)^{2}}{2} - \frac{10 (10 + 1) (2 \times 10 + 1)}{6})$
$= 11 - 2 \times 10^{2} \times 11^{2} + 2 \times 10 \times 11 \times 7$
$= 11 - 220 (110 - 7)$
$= 11 - 220 (103)$
$∴ α = 11, β = 103$

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If 1+(1−22⋅1)+(1−42⋅3)+(1−62⋅5)+⋯+(1−202⋅19)=α−220β, then an ordered pair (α,β) is equal to

If $1 + (1 - 2^{2} \cdot 1) + (1 - 4^{2} \cdot 3) + (1 - 6^{2} \cdot 5) + \dots + (1 - 20^{2} \cdot 19) =$ $α - 220 β$ , then an ordered pair $(α, β)$ is equal to