Geometric Sequences: Multiplying by the Same Factor

A geometric sequence multiplies by the same number every time.

2, 6, 18, 54, 162, ...

Each term is 3 times the one before. That constant multiplier—the common ratio—defines everything. Arithmetic sequences add; geometric sequences multiply.

This is the pattern of exponential change. Growth that feeds on itself. Decay that accelerates as it shrinks. The math of compound interest, population explosions, and radioactive half-lives.

The Formula

If the first term is a₁ and the common ratio is r, then the nth term is:

aₙ = a₁ · rⁿ⁻¹

Why n - 1? Because to get from the first term to the nth term, you multiply by r exactly n - 1 times.

• 1st term: a₁ (no multiplications)
• 2nd term: a₁ · r (one multiplication)
• 3rd term: a₁ · r² (two multiplications)
• nth term: a₁ · rⁿ⁻¹

For the sequence 2, 6, 18, 54, ... (a₁ = 2, r = 3):

The 10th term? a₁₀ = 2 · 3⁹ = 2 · 19683 = 39,366.

Geometric sequences grow fast. By the 10th term, we've gone from 2 to nearly 40,000.

Finding the Common Ratio

The common ratio is the quotient of consecutive terms:

r = aₙ₊₁ / aₙ

For 5, 20, 80, 320, ..., the ratio is 20/5 = 4.

You can also find r from any two terms if you know their positions:

r = (aₘ / aₙ)^(1/(m-n))

If the 2nd term is 6 and the 5th term is 162:

r = (162/6)^(1/3) = 27^(1/3) = 3

Three positions apart, the ratio of terms is 27, so the common ratio is the cube root: 3.

The Geometric Mean

The geometric mean between two positive numbers is the number that makes a geometric sequence with them.

What's the geometric mean of 4 and 16? The number x such that 4, x, 16 is geometric.

For this to be geometric, we need x/4 = 16/x. Solving: x² = 64, so x = 8.

Check: 4, 8, 16 has ratio 2. It works.

The geometric mean of a and b is √(ab). It's what we use when ratios matter more than differences.

Three Behaviors Based on r

Geometric sequences behave completely differently depending on the common ratio:

|r| > 1: Explosive growth (or decay)

2, 6, 18, 54, ... (r = 3) → terms grow without bound

The sequence diverges. Each term is bigger than the last by an increasing amount.

|r| < 1: Decay toward zero

16, 8, 4, 2, 1, 0.5, 0.25, ... (r = 0.5) → terms shrink toward 0

The sequence converges to 0. Each term is smaller than the last, and they approach zero.

|r| = 1: Constant or oscillating

5, 5, 5, 5, ... (r = 1) → stays constant 5, -5, 5, -5, ... (r = -1) → oscillates forever

This is the critical insight: Whether a geometric sequence converges depends entirely on whether |r| < 1.

Negative Common Ratios

When r is negative, the sequence alternates sign:

3, -6, 12, -24, 48, ... (r = -2)

The terms flip between positive and negative while growing in magnitude.

If |r| < 1 with r negative, the terms still converge to 0, but they oscillate on the way:

1, -0.5, 0.25, -0.125, 0.0625, ... (r = -0.5)

The terms get closer to zero while bouncing above and below it.

Geometric vs. Arithmetic

The difference between arithmetic and geometric sequences is the difference between linear and exponential:

A salary with $2,000 annual raises is arithmetic. A salary that increases 5% annually is geometric.

Early on, arithmetic and geometric sequences can look similar. But geometric always wins eventually (if |r| > 1). Exponential growth eventually outpaces any linear growth, no matter how steep.

Half-Lives and Doubling Times

Geometric sequences naturally model processes with half-lives or doubling times.

Half-life: How long until half remains. If r = 0.5 per unit time, each step halves the amount.

A radioactive sample with a 10-year half-life:

• Year 0: 100 g
• Year 10: 50 g
• Year 20: 25 g
• Year 30: 12.5 g

Doubling time: How long until the amount doubles. If r = 2 per unit time, each step doubles.

Bacteria doubling every hour:

• Hour 0: 1 cell
• Hour 1: 2 cells
• Hour 2: 4 cells
• Hour 10: 1,024 cells
• Hour 20: 1,048,576 cells

This is why exponential growth is so dangerous—and so powerful. Ten doublings gives you a thousand. Twenty gives you a million.

Compound Interest Is Geometric

When money earns interest, each period's interest is calculated on the current balance—including previous interest.

$1,000 at 5% annual interest:

• Year 0: $1,000.00
• Year 1: $1,050.00
• Year 2: $1,102.50
• Year 3: $1,157.63

This is a geometric sequence with r = 1.05.

After n years: A = 1000 · (1.05)ⁿ

Compound interest is why time matters so much in investing. The geometric sequence doesn't care about the starting point as much as the number of multiplications.

Finding Terms and Positions

Finding the nth term: Direct substitution.

For 3, 12, 48, ... (a₁ = 3, r = 4), the 6th term is: a₆ = 3 · 4⁵ = 3 · 1024 = 3,072

Finding which term equals a value: Solve the exponential equation.

Which term of 2, 6, 18, ... equals 1,458?

2 · 3ⁿ⁻¹ = 1,458 3ⁿ⁻¹ = 729 3ⁿ⁻¹ = 3⁶ n - 1 = 6 n = 7

The 7th term is 1,458.

This requires recognizing 729 as a power of 3, or using logarithms for messier numbers.

The Limit of a Geometric Sequence

Where does aₙ = a₁ · rⁿ⁻¹ go as n → ∞?

• If |r| < 1: aₙ → 0 (converges to zero)
• If |r| > 1: aₙ → ±∞ (diverges)
• If r = 1: aₙ = a₁ (constant)
• If r = -1: aₙ oscillates (diverges)

The magic of |r| < 1: The terms get smaller fast enough that even an infinite number of them can sum to something finite.

This is why geometric series are so important. Unlike arithmetic series (which always diverge if you keep adding), geometric series can converge—if the ratio is small enough.

Why Geometric Sequences Matter

1. They model multiplicative processes. Anything that grows by percentages—interest, populations, decay—follows geometric patterns.
2. They connect to exponential functions. The continuous analog of a geometric sequence is the exponential function y = a · bˣ.
3. Their series can converge. This is the key fact. Infinite geometric series have finite sums when |r| < 1. This makes calculus possible.
4. They're everywhere in nature and finance. From radioactive decay to viral spread to compound interest, geometric sequences describe how change compounds.

The geometric sequence says: when growth feeds on itself, things accelerate. Small ratios mean gradual decay. Large ratios mean explosion. And the magic ratio |r| < 1 is where infinity becomes finite.

Part 3 of the Sequences Series series.

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