Solve Series S_n: 1056, 1088, 1120, 1332

[klen]

S_n = \sum_{k=1}^ {4n} {(-1)^\frac{k(k+1)}{2}} {k^2}.

Then Sn can take what values among:
1056, 1088, 1120, 1332

 

[tiny-tim]

hi klen!

simplest way is probably to do the sum for 4m-3,4m-2,4m-1,4m (only) first …

what do you get? ​

 

[HallsofIvy]

S_n = \sum_{k=1}^ {4n} {(-1)^\frac{k(k+1)}{2}} {k^2}.

Then Sn can take what values among:
1056, 1088, 1120, 1332

Did you, for example, notice that every time n increases by 1, the sum has 4 new terms, two negative and two positive? Do you notice that for n= 1, so there are 4 terms, the sum is 20= 4(5) and when n= 2, so there are 8 terms, the sum is 72= 8(9)?