`Let S_n=underset(k=1)overset(4n)Sigma (-1)^((k(k+1))/2)k^2`.Then `S_n` can take value (s)

Options

A. 1056
B. 1088
C. 1120
D. 1332

Answer

A, D

Explanation

`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.