The formula for the sum of a finite geometric serious with first term a and common ratio r≠1, is ∑i=1nari−1=a+ar+ar2+⋯+arn−1=a(rn−1)r−1.

The sum ∑i=1n(2i+2i) is simplified as ∑i=1n2i+∑i=1n22i−1.

Here, the sum ∑i=1n22i−1 is in the form ∑i=1nari−1 where, a=2 and r=2.

By the above formula, the value of the sum ∑i=1n22i−1 is simplified as,

∑i=1n22i−1=2[2n−1]2−1=2(2n−1)

Thus, the value of the sum ∑i=1n22i−1 is 2(2n−1).

Simplify the expression ∑i=1n(2i+2i) and obtain the value of the sum.

∑i=1n(2i+2i)=∑i=1n2i+∑i=1n22i−1=2n(n+1)2+2(2n−1)=n(n+1)+2(2n−1)=2n+1+n2+n−2

Thus, the value of the sum ∑i=1n(2i+2i) is 2n+1+n2+n−2.