Question

What does ∑ k=0 to n of ar^k equal?

A. ar^n.
B. (ar^n - 1) / (r - 1).
C. a + ar + ar^2 + ... + ar^n.
D. a(1 - r^n) / (1 - r).

Answer 1

The sum of a finite geometric series from k=0 to n of ar^k equals a(1 - r^n)/(1 - r), so the answer is D. a(1 - r^n) / (1 - r).

Step-by-step explanation:

The sum ∑ from k=0 to n of ar^k equals a geometric series with n terms. The formula for the sum of a finite geometric series is given by a(1-r^n) / (1-r) when |r| < 1 (where a is the first term, r is the common ratio, and n is the number of terms). If r = 1, then the sum simply becomes an, but this is a special case and not the focus of this question.

So, the correct answer to 'What does ∑ k=0 to n of ar^k equal?' is D. a(1 - r^n) / (1 - r). This is because this formula represents the sum of a finite geometric series.