`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 and 1332
B. 1056 and 1088
C. 1120 and 1332
D. 1332 and 1432

Answer

A

Explanation

We have `S_(n)=underset(k=1)overset(4n)sum(-1)^((k(k+1))/(2))k^(2)`
`rArr" "S_(n)=-1^(2)-2^(2)+3^(2)+4^(2)-5^(2)-6^(2)+7^(2)+8^(2). . . .` upto 4n terms
`rArr" "S_(n)=(3^(2)-1^(2))+(4^(2)-2^(2))+(7^(2)-5^(2))+(8^(2)-6^(2))+ . . .. ` up to 2n terms
`rArr" "S_(n)=2(4+6+12+14+20+22 . . . .` up to 2n terms
`rArr" "S_(n)=2{underset("n-terms")(4+12+20+ . . .)+underset("n-terms")(6+14+22+ . . . )}`
`rArr" "S_(n)=2{:[(n)/(2){8+(n-1)xx8}+(n)/(2){12+(n-1)xx8}]:}`
`rArr" "S_(n)=8n^(2)+8n^(2)+4n`
`rArr" "S_(n)=4n(4n+1)`
`rArr" "S_(n)`= Product of a multiple of 4 and its successor.
Clearly, `1056=32xx33=(4xx8)((4xx8)+1)and1332=326xx37=(4xx9)+1)` are products of multiple of 4 and its successor.
Hence, `S_(n)` can take values 1056 and 1332.