let a_i = 2^i

Question

20
Find Σ a_(i+1)-a_i
i=1

Answer 1

Note the sum. You have

a2-a1 + a3-a2 + a4-a3 + ... + a21-a20
= a21-a1
= 2^21 - 2^1

There's really no reason to write the sum as you did. Since you know a_i explicitly, it would be better just to say

20
Σ 2^(i+1)-2^i
i=1