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

Let `S_(n)=sum_(k=1)^(4n)(-1)^((k(k+1))/2)k^(2)`. Then `S_(n)` can take values

A

1056

B

1088

C

1120

D

1232

Text Solution

Verified by Experts

The correct Answer is:
a
`S_(n) = sum_(k=1)^(4n) (-1)^((k(k+1))/2).k^(2)`
` = (-2)^(2) - 2^(2) +3^(2) +4^(2) - 5^(2) -6^(2) +7^(2) +8^(2)-…`
` (3^(2)-1^(2))+(4^(2)-2^(2))+(7^(2)-5^(2))+(8^(2)-6^(2))+...`
`=2underset("n terms")(ubrace("{(4+12+...)"))+underset("n terms")(ubrace("(6+14+22+...)}"))`
` = 2 [n/2 {2xx 4+(n-1)8}+n/2{2xx6 +(n-1)8}]`
` =2 [n (4+4n-4)+n(6+4n-4)]`
` =2 [ 4n^(2) +4n^(2) +2n] = 4n (4n+1)`
Here , `1056 = 32 xx 33, 1088 = 32 xx34`
` 1120 = 32 xx35 ,1332 = 36 xx37`
Hence , 1056 and 1332 are possible answers .
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-SEQUENCES AND SERIES -EXERCISE 20
  1. Let S(n)=sum(k=1)^(4n)(-1)^((k(k+1))/2)k^(2). Then S(n) can take value...

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