What is the sum of the first 12 terms of the geometric series below?
one-half plus 2 plus 8 plus 32 plus ellipsis
(1 point)
Responses
The sum of the first 12 terms of the given geometric series can be found using the formula for the sum of a geometric series:
S_n = a(1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term (a) is 1/2 and the common ratio (r) is 4. We want to find the sum of the first 12 terms, so n = 12.
Plugging these values into the formula, we get:
S_12 = (1/2)(1 - 4^12) / (1 - 4)
Simplifying this expression gives:
S_12 = (1/2)(1 - 16^2) / (1 - 4)
= (1/2)(1 - 256) / (1 - 4)
= (1/2)(-255) / (-3)
= (1/2)(85)
= 85/2
= 42.5
So, the sum of the first 12 terms of the geometric series is 42.5.