Mathematics
Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms (e.g. 3, 6, 12, 24, ... where r = 2). The nth term is a_n = a_1·r^(n-1). Finite sums use S_n = a_1·(1 - r^n)/(1 - r); infinite sums converge when |r| < 1 to a_1/(1 - r).
How to Approach Geometric Sequences
- 01
Find the common ratio r
Divide any term by the previous one. If the ratio is constant, it's geometric. If r > 1 the sequence grows; if 0 < r < 1 it shrinks; if r < 0 it alternates signs.
- 02
Apply the nth term formula
a_n = a_1·r^(n-1). Plug in the first term, the ratio, and which term you want. Note: r is raised to (n-1), not n.
- 03
Sum carefully
Finite: S_n = a_1·(1 - r^n)/(1 - r). Infinite (|r| < 1): S = a_1/(1 - r). If |r| ≥ 1, the infinite series diverges — no finite sum.
Frequently Asked Questions
When does an infinite geometric series converge?
When |r| < 1. The terms get smaller and smaller, summing to a finite limit a_1/(1 - r). If |r| ≥ 1 the series grows without bound.
What's an example of an infinite geometric sum?
0.999... = 9/10 + 9/100 + 9/1000 + ... = (9/10)/(1 - 1/10) = 1. The infinite decimal exactly equals 1.
How is this used in real life?
Compound interest, exponential decay (radioactive, bacterial), Zeno's paradoxes, fractal geometry, signal processing, present-value finance.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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