InterviewSolution
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InterviewSolution
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Let `S_(n)=sum_(k=1)^(4n) (-1)^((k(k+1))/(2))k^(2)`. Then, `S_(n)` can take the value (s)A. 1056 and 1332B. 1056 and 1088C. 1120 and 1332D. 1332 and 1432 |
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Answer» Correct Answer - A 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. |
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