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

embizze | Certified Educator

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` .