Geometric Sequence

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

Description

A geometric sequence (or progression) is a list of numbers where each term is found by multiplying the previous term by a constant value called the **common ratio ($r$)**.

The formula $a_n = a_1 \cdot r^{n-1}$ allows you to calculate any term in the sequence without finding all the previous ones. * $a_1$ is the first term. * $r$ is the multiplier (ratio). * $n-1$ is the number of multiplications applied to the start.

This behavior describes exponential growth (if $r > 1$) or exponential decay (if $0 < r < 1$).

History & Origins

Geometric progressions appear in some of the oldest mathematical stories. The Wheat and Chessboard Problem: A famous legend from India involves a wise man who asked for 1 grain of wheat on the first square of a chessboard, 2 on the second, 4 on the third, and so on. This is a geometric sequence with $r=2$. The total amount of wheat (a geometric series) would exceed global production for centuries. Euclid (c. 300 BC): Discussed geometric progressions in Elements, Book IX.

Derivation by Iteration

We can find the pattern by writing out the terms step-by-step.

Term 1: $a_1$

Term 2: $a_2 = a_1 \cdot r$

Term 3: $a_3 = a_2 \cdot r = (a_1 \cdot r) \cdot r = a_1 \cdot r^2$

Term 4: $a_4 = a_3 \cdot r = (a_1 \cdot r^2) \cdot r = a_1 \cdot r^3$

Pattern: The exponent of $r$ is always one less than the term number ($n$).

General formula: $a_n = a_1 \cdot r^{n-1}$.

Variables

Symbol	Meaning
`aₙ`	The nth term
`a₁`	First term
`n`	Position of the term
`r`	Common ratio (Multiplier)

Examples

Basic Calculation

Problem: Find the 5th term of the sequence 2, 6, 18, 54...

Solution:

a_5 = 2 * 3^(5-1) = 2 * 81 = 162

Bacteria Growth

Problem: A bacteria culture starts with 100 cells and doubles every hour. How many bacteria are there at the start of the 6th hour?

Solution: 3,200

Identify sequence: 100, 200, 400...
First term $a_1 = 100$. Ratio $r = 2$.
We want the 6th term ($n=6$).
Formula: $a_6 = 100 \cdot 2^{6-1} = 100 \cdot 2^5$.
Calculate: $100 \cdot 32 = 3,200$ bacteria.

Car Depreciation

Problem: A car is purchased for $20,000. It retains 85% of its value each year (depreciates by 15%). What is its value in year 5?

Solution: $10,440

Start ($a_1$): $20,000$. Ratio $r = 0.85$.
Note: Year 1 is start. Year 5 is the 5th term.
Formula: $a_5 = 20,000 \cdot (0.85)^{5-1} = 20,000 \cdot (0.85)^4$.
Calculate: $20,000 \cdot 0.522 = 10,440$.

Common Mistakes

❌ Mistake

Using n instead of n-1

✅ Correction

A common error is calculating $a_1 \cdot r^n$. This effectively skips the first term or shifts the sequence. Remember the first term has 0 multiplications.

❌ Mistake

Confusing Arithmetic and Geometric

✅ Correction

Check the pattern. If you ADD to get the next number, it's Arithmetic. If you MULTIPLY, it's Geometric.

Real-World Applications

Finance: Compound Interest

Compound interest is a geometric sequence. If you invest money at 5% interest, your money is multiplied by 1.05 each year. $A = P(1+r)^t$ is just the geometric sequence formula rewritten.

Music: Octaves

Musical notes follow a geometric progression. The frequency of an "A" note is 440 Hz. The next "A" (one octave up) is 880 Hz ($r=2$). The 12 semitones in between form a geometric sequence with ratio $\sqrt[12]{2}$.

Frequently Asked Questions

Can r be negative?

Yes. This creates an alternating sequence where signs flip (e.g., 2, -4, 8, -16...).

What if r is less than 1?

The sequence decreases (exponential decay), approaching zero. Example: 100, 50, 25...