Geometric Sequences and Series

Core Idea

A geometric sequence has a constant ratio r between consecutive terms: a_n = a_1 * r^n-1. The sum of the first n terms (geometric series) is S_n = a_1 * (1 - rⁿ)/(1 - r) for r != 1. If |r| < 1, the infinite geometric series converges to S = a_1/(1 - r). Geometric sequences model exponential growth and decay. The infinite series formula is foundational for calculus and finance.

How It's Best Learned

Identify common ratios in sequences. Derive the finite sum formula by multiplying S_n by r and subtracting. Explore the infinite series by examining what happens as n grows when |r| < 1. Apply to compound interest, bouncing balls, and repeating decimals as infinite geometric series.

Common Misconceptions

• Confusing common difference (arithmetic) with common ratio (geometric).
• Using the sum formula when r = 1 (division by zero; the sum is simply n*a_1).
• Applying the infinite series formula when |r| >= 1 (the series diverges).
• Sign errors in the sum formula when r is negative.

Explainer

You already understand exponential functions: y = a · r^x, where a is the starting value and r is the growth (or decay) factor applied repeatedly. A geometric sequence is the discrete version of exactly this idea. Instead of a continuous curve, you have a list: a₁, a₁r, a₁r², a₁r³, …. Each term is r times the previous one. The number r is the common ratio — the defining characteristic that makes a sequence geometric. To check whether a sequence is geometric, divide any term by the one before it: if you always get the same number, it is geometric.

The nth term formula a_n = a₁ · rⁿ⁻¹ follows directly from the exponential structure. Starting at a₁ (when n = 1, r⁰ = 1), each successive step multiplies by r. This connects to every exponential model you have seen: compound interest starts with a principal and multiplies by (1 + rate) each period; radioactive decay multiplies by a fraction each half-life; a bouncing ball reaches r times the previous height on each bounce. All of these are geometric sequences. Whether |r| > 1 (growth), 0 < |r| < 1 (decay), or r < 0 (alternating sign) determines the sequence's long-run behavior.

The finite sum formula S_n = a₁(1 − rⁿ)/(1 − r) is derived by an elegant algebraic trick. Write S_n = a₁ + a₁r + a₁r² + … + a₁rⁿ⁻¹. Multiply both sides by r: rS_n = a₁r + a₁r² + … + a₁rⁿ. Subtract: S_n − rS_n = a₁ − a₁rⁿ, so S_n(1 − r) = a₁(1 − rⁿ), giving the formula. This "multiply and subtract" technique is a foundational algebraic strategy that reappears in many contexts, including proving limits and summing other special series.

The infinite series formula S = a₁/(1 − r) for |r| < 1 emerges from taking the limit as n → ∞. When |r| < 1, the term rⁿ shrinks toward zero, so the factor (1 − rⁿ) in the numerator approaches 1, leaving a₁/(1 − r). The series converges because each additional term contributes a shrinking fraction of the last, and the cumulative total is bounded. A striking application: the repeating decimal 0.333… = 3/10 + 3/100 + 3/1000 + … is an infinite geometric series with a₁ = 3/10 and r = 1/10. The formula gives S = (3/10)/(1 − 1/10) = (3/10)/(9/10) = 1/3 exactly — confirming that 0.333… and 1/3 are the same number.