Question

Find the value of the series

Answer 1

Step-by-step explanation:

that is

sum(2^r) for r=1 to n, plus sum(1/2) for r=1 to n.

and that is

sum(2^r) + n/2 for r=1 to n.

2^r is a geometric sequence with 2 being the common ratio (every new term is created by multiplying the previous term by 2).

and since r is starting at 1, the first term a1 = 2.

the formula for the sum of a finite geometric sequence is

Sn = a1×(1 - r^n) / (1 - r)

with r being the common ratio .

so, in our case

Sn = 2×(1 - 2^n) / (1 - 2)

Sn = (2 - 2^(n+1)) / -1 = 2^(n+1) - 2

and so, in total we get

2^(n+1) - 2 + n/2 = 2^(n+1) + (n - 4)/2