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Let S(n)=sum(k=1)^(4n)(-1)^((k(k+1))/(2)...

Let `S_(n)=sum_(k=1)^(4n)(-1)^((k(k+1))/(2))*k^(2)`, then `S_(n)` can take value

A

1056

B

1088

C

1120

D

1332

Text Solution

Verified by Experts

The correct Answer is:
A, D
`S_(n)=-1^(2)-2^(2)+3^(2)+4^(2)-5^(2)-6^(2)+7^(2)+8^(2)-"........"+(4n-1)^(2)+(4n)^(2)`
`=(3^(2)-1^(2))+(4^(2)-2^(2))+(7^(2)-5^(2))+(8^(2)-6^(2))+"........"+[{(4n-1)^(2)-(4n-3)^(2)}+{(4n)^(2)-(4n-2)^(2)}]`
`=4[2+3+6+7+10+11+"......"+(4n-2)+(4n-1)]`
`=8{(1+3+5+"......"+(2n-1)}+4{3+7+11+"........."+(4n-1)}`
`=16n^(2)+4n=4n(4n+1),n inN`
Satisfied by a and d where n=8,9, respectively.
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