Question

What is the sum of `1, 4, 16, 64, ..., a_10`?

Answer 1

`349525`

Explanation:

This is a geometric sequence with initial term `1` and common ratio `4`:

`a_1 = 1`

`a_n = a_1 * 4^(n-1) = 4^(n-1)`

`sum_(n=1)^10 4^(n-1) = 1/3(4-1) sum_(n=1)^10 4^(n-1)`

`=1/3(4 sum_(n=1)^10 4^(n-1) - sum_(n=1)^10 4^(n-1))`

`=1/3(sum_(n=2)^11 4^(n-1) - sum_(n=1)^10 4^(n-1))`

`=1/3(4^10 + color(red)(cancel(color(black)(sum_(n=2)^10 4^(n-1)))) - 1 - color(red)(cancel(color(black)(sum_(n=2)^10 4^(n-1)))))`

`=1/3(4^10-1)`

`=1/3(2^20-1)`

`=1/3(1048576-1)`

`=1048575/3`

`= 349525`