Sequences & Series

Geometric Sequences

Writing explicit and recursive formulas for a geometric sequence.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Arithmetic Sequences

A Sequence That Grows by Multiplying

3, 6, 12, 24, .... Before reading on, notice this pattern is fundamentally different from an arithmetic sequence — predict the rule connecting consecutive terms, and think about which earlier function type in this course behaves the same way.

Definition — Geometric Sequence

A sequence where each term is found by multiplying the previous one by a constant, called the common ratio, r. The explicit formula is aₙ = a₁ · r^(n−1).

This is the exact same structure as an exponential function, y = a · bˣ, restricted to whole-number inputs — the starting term a₁ plays the role of a, and the common ratio r plays the role of b. Just as an exponential function's compounding growth eventually outpaces linear growth, a geometric sequence's terms eventually outpace an arithmetic sequence's, for exactly the same underlying reason: multiplicative growth compounds on itself.

Worked Example — Finding an Explicit Formula

Find an explicit formula for 3, 6, 12, 24, .... The first term is a₁ = 3, and the common ratio is r = 2 (each term is double the last). The explicit formula: aₙ = 3 · 2^(n−1).

Worked Example — Finding a Specific Term

Using aₙ = 3 · 2^(n−1), find the 10th term. a₁₀ = 3 · 2⁹ = 3 · 512 = 1,536.

Worked Example — Finding the Common Ratio from Two Non-Adjacent Terms

A geometric sequence has a₂ = 20 and a₅ = 2,500. Find r. Since a₅ = a₂ · r³ (three multiplications by r to get from the 2nd term to the 5th), 2500 = 20 · r³, so r³ = 125, giving r = 5.

Tip

Test whether a sequence is geometric by dividing consecutive terms rather than subtracting them — a constant quotient signals geometric, the way a constant difference signals arithmetic.

Common Mistakes

Subtracting consecutive terms to find a common ratio, treating a geometric sequence like an arithmetic one.
A geometric sequence's terms share a constant ratio, found by dividing consecutive terms, not a constant difference found by subtracting them.
Miscounting the number of multiplications by r between two non-adjacent terms, such as using r⁴ instead of r³ between the 2nd and 5th terms.
Count the gap between term positions carefully — going from the 2nd term to the 5th term takes exactly 5 − 2 = 3 multiplications by r, not 4.

Key Takeaways

A geometric sequence multiplies by the same common ratio at every step.
Its explicit formula, aₙ = a₁ · r^(n−1), mirrors an exponential function's structure, restricted to whole-number inputs.
A constant ratio (not a constant difference) between consecutive terms identifies a sequence as geometric.

Summary

Geometric sequences are exponential functions living on whole numbers, the discrete counterpart to arithmetic sequences and linear functions. The final lesson in this unit finds the sum of many terms in a sequence at once.

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