Unit 2: Exponential & Logarithmic Functions

Geometric Sequences

Connecting geometric sequences to exponential growth and decay.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

When Ratios Stay Constant Instead of Differences

Constant differences in a table exposed a polynomial's degree. Before reading on: what would it mean, instead, if the ratio between consecutive outputs in a table stayed constant?

Definition — Geometric Sequence

A geometric sequence has a constant ratio r between consecutive terms: aₙ = a₁ · r ⁿ⁻¹, where a₁ is the first term. Each term is the previous term multiplied by the same factor r, the multiplicative counterpart to an arithmetic sequence's constant added difference.

a_n = a_1 \cdot r^{\,n-1}

A geometric sequence is exactly what you get from sampling an exponential function f(x) = a · bˣ at whole-number inputs — the constant ratio r is just the base b. That's why exponential functions have their own version of the constant-differences test from Unit 1: instead of constant differences over equal-width input intervals, exponential functions have constant ratios over equal-width input intervals. Multiplication takes the place addition held for polynomials.

Worked Example — Finding a Geometric Sequence's Formula

The sequence 3, 6, 12, 24, 48 has ratio r = 6/3 = 2 (check: 12/6 = 2, 24/12 = 2, 48/24 = 2 — constant). With a₁ = 3, the explicit formula is aₙ = 3 · 2ⁿ⁻¹. Verify the 5th term: a₅ = 3 · 2⁴ = 3 · 16 = 48, matching the given sequence.

Worked Example — Confirming a Constant Ratio from a Function

For f(x) = 5 · 3ˣ, evaluate at x = 0, 1, 2, 3: f(0) = 5, f(1) = 15, f(2) = 45, f(3) = 135. The ratios between consecutive outputs are 15/5 = 3, 45/15 = 3, and 135/45 = 3 — constant, and equal to the base, 3. A ratio r above 1 corresponds to a growth rate of (r − 1); here r = 3 means the output triples — a 200% increase — every time x increases by 1.

Tip

To test whether a table of values could come from an exponential function, divide consecutive outputs instead of subtracting them — a constant quotient is the exponential signature, the multiplicative twin of a polynomial's constant difference.

Common Mistakes

Subtracting consecutive terms of a geometric sequence and expecting a constant difference.
A geometric sequence's differences generally aren't constant — only its ratios are. Divide, don't subtract, when testing for a geometric pattern.
Reading a ratio like r = 0.85 as '85% growth' instead of '15% decay.'
A ratio below 1 signals decay: r = 1 − (decay rate), so r = 0.85 means the quantity loses 15% each step, not gains 85%.

Key Takeaways

A geometric sequence has a constant ratio r between consecutive terms: aₙ = a₁ · rⁿ⁻¹.
Sampling an exponential function f(x) = a · bˣ at whole-number inputs produces a geometric sequence with ratio r = b.
Constant ratios over equal-width input intervals are the exponential-function counterpart to a polynomial's constant differences.

Summary

Geometric sequences are exponential functions viewed one step at a time. The next lesson builds a full exponential model from a growth rate or a pair of data points, the same way a sequence's ratio and first term build its explicit formula.

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