Q: When does a geometric series converge?

A geometric series converges to a finite value only if the absolute value of the common ratio is strictly less than 1, that is |r| < 1. In this case, the infinite sum is S∞ = a1 / (1 − r). If |r| ≥ 1, the partial sums grow without bound in magnitude or oscillate and there is no finite limit.

Q: What is the difference between arithmetic and geometric sequences?

In an arithmetic sequence each term is obtained by adding a constant difference d to the previous term, whereas in a geometric sequence each term is obtained by multiplying the previous term by a constant ratio r. The general term of an arithmetic sequence is an = a1 + (n−1)d, while for a geometric sequence it is an = a1 · r^(n−1).

Question 1

What is a geometric sequence?

Accepted Answer

A geometric sequence (or geometric progression) is a sequence of numbers where each term is obtained by multiplying the previous term by a constant called the common ratio r. If the first term is a1, then the nth term is given by an = a1 · r^(n−1).

Question 2

What does this geometric sequence calculator do?

Accepted Answer

This calculator generates terms of a geometric sequence, finds the nth term, computes the sum of the first n terms, checks convergence, and can solve for an unknown among first term, common ratio, index n or nth term using the standard geometric progression formulas.

Question 3

How do you find the nth term of a geometric sequence?

Accepted Answer

If a1 is the first term and r is the common ratio, the nth term is an = a1 · r^(n−1). For example, for a1 = 3 and r = 2, the 5th term is a5 = 3 · 2^(5−1) = 3 · 16 = 48.

Question 4

What is the formula for the sum of a geometric sequence?

Accepted Answer

For a geometric sequence with first term a1, common ratio r and n terms, the finite sum is Sn = a1 · (1 − r^n) / (1 − r) when r ≠ 1. If r = 1, then Sn = n · a1. When |r| < 1 and n tends to infinity, the infinite sum is S∞ = a1 / (1 − r).

Question 5

When does a geometric series converge?

Accepted Answer

A geometric series converges to a finite value only if the absolute value of the common ratio is strictly less than 1, that is |r| < 1. In this case, the infinite sum is S∞ = a1 / (1 − r). If |r| ≥ 1, the partial sums grow without bound in magnitude or oscillate and there is no finite limit.

Question 6

What is the difference between arithmetic and geometric sequences?

Accepted Answer

In an arithmetic sequence each term is obtained by adding a constant difference d to the previous term, whereas in a geometric sequence each term is obtained by multiplying the previous term by a constant ratio r. The general term of an arithmetic sequence is an = a1 + (n−1)d, while for a geometric sequence it is an = a1 · r^(n−1).