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:

$$S = \frac{a\left(1\right)\cdot \left(1-{r}^{n}\right)}{1-r}$$

Here n counts terms, so the power is ${r}^{n}$, not ${r}^{n-1}$. Check it: $\frac{1\cdot \left(1-{4}^{4}\right)}{1-4}$ 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.