# 1 - 4 + 9 - 16 + 25 - 36 +... -64 =_________Is there any formualr to do this sequence?

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### 1 Answer

Find the sum of 1-4+9-16+25-...-64:

I am unaware of a general formula to find the partial sum of an alternating series. There are formulas for special cases such as alternating harmonic series, alternating geometric series, etc...

**For this case there is a formula:** `sum_(i=1)^n(-1)^(n-1)n^2=(-1)^(n-1)((n(n+1))/2)`

For the given case n=8 so the sum is `(-1)^(7)((8*9)/2)=-36`

Here are the first few terms:

`n` `S_n`

1 `(-1)^0((1*2)/2)=1`

2 `(-1)^1((2*3)/2)=-3` which is 1-4

3 `(-1)^2((3*4)/2)=6` which is 1-4+9

etc...

You can see some of the derivation by considering different groupings:

(1-4)+(9-16)+(25-36)+(49-64)=-3+-7+-11+-15; the partial sums are -3,-10,-21,-36. You might recognize these as every other triangular number.

Another possible grouping: 1+(-4+9)+(-16+25)+(-36+49)+(-64)=1+5+9+13+... the partial sums of which are 1,6,15,28 which are the rest of the triangular numbers.

The formula for the nth triangular number is `(n(n+1))/2` .