Geometric Sums

Add 1 + 4 + 16 + 64 with a trick. Call the sum S, then multiply every term by the ratio 4: 4S = 4 + 16 + 64 + 256. The two lists share 4, 16 and 64, so subtracting S from 4S cancels them: 4S - S = 256 - 1. That is 3S = 255, so S is 85.

The same subtraction works for any geometric sequence and gives the sum of its first n terms:

S = a(1) × (1 - r^n)/(1 - r)

Here n counts terms, so the power is r^n, not r^(n - 1). Check it: 1 × (1 - 4^4)/(1 - 4) is -255 over -3, which is 85. The formula breaks only when r is 1, where its bottom is 0; then every term is the same and the sum is n times the first.