Geometric Sequences- Definition, Formulas, and Examples
What Is a Geometric Sequence?
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number. That fixed number is called the common ratio.
It's simple. You start with one number, then keep multiplying by the same value over and over.
Example: 2, 6, 18, 54, 162...
Here, each term multiplies by 3. That's the common ratio.
The Geometric Sequence Formula
The nth term of a geometric sequence follows this rule:
an = a1 × r(n-1)
Where:
- an = the nth term you want to find
- a1 = the first term in the sequence
- r = the common ratio
- n = the term number
That's it. Plug in your values and solve.
How to Find the Common Ratio
The common ratio is found by dividing any term by the term before it:
r = an ÷ a(n-1)
Using the example 2, 6, 18, 54:
- 6 ÷ 2 = 3
- 18 ÷ 6 = 3
- 54 ÷ 18 = 3
The common ratio is 3. It stays the same throughout the sequence.
Geometric Sequence vs Arithmetic Sequence
Don't confuse these two. They work differently:
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Operation | Addition or subtraction | Multiplication or division |
| Constant term | Common difference (d) | Common ratio (r) |
| Example | 5, 10, 15, 20... | 5, 10, 20, 40... |
| Pattern | Adds 5 each time | Multiplies by 2 each time |
If the numbers grow by adding a constant, it's arithmetic. If they grow by multiplying by a constant, it's geometric.
Examples of Geometric Sequences
Example 1: Positive Common Ratio
Sequence: 3, 9, 27, 81, 243
Common ratio: 9 ÷ 3 = 3
Find the 6th term:
an = a1 × r(n-1)
a6 = 3 × 3(6-1) = 3 × 35 = 3 × 243 = 729
Example 2: Fractional Common Ratio
Sequence: 100, 50, 25, 12.5
Common ratio: 50 ÷ 100 = 0.5 (or 1/2)
Find the 5th term:
a5 = 100 × (0.5)(5-1) = 100 × (0.5)4 = 100 × 0.0625 = 6.25
Example 3: Negative Common Ratio
Sequence: 4, -8, 16, -32
Common ratio: -8 ÷ 4 = -2
Find the 5th term:
a5 = 4 × (-2)(5-1) = 4 × (-2)4 = 4 × 16 = 64
The terms alternate between positive and negative when r is negative.
How to Identify a Geometric Sequence
Quick checklist:
- Pick any two consecutive terms
- Divide the second by the first
- If you get the same answer every time, it's geometric
If the ratios keep changing, it's not a geometric sequence.
Sum of a Geometric Sequence
Sometimes you need the sum of terms, not just the terms themselves. Use this formula for the sum of the first n terms:
Sn = a1 × (1 - rn) ÷ (1 - r)
This works when r ≠ 1.
Sum Example
Find the sum of the first 5 terms of: 2, 6, 18, 54, 162
a1 = 2, r = 3, n = 5
S5 = 2 × (1 - 35) ÷ (1 - 3)
S5 = 2 × (1 - 243) ÷ (-2)
S5 = 2 × (-242) ÷ (-2) = 242
Check: 2 + 6 + 18 + 54 + 162 = 242 ✓
Practical Applications
Geometric sequences show up in real situations:
- Compound interest — money grows by multiplying each period
- Population growth — bacteria doubling each hour
- Physics — depreciation of value over time
- Computer science — algorithm complexity analysis
When something grows or shrinks by a consistent percentage, you're looking at a geometric sequence.
Getting Started: Solve Any Geometric Sequence Problem
Step 1: Identify a1 (first term)
Step 2: Find r by dividing one term by the previous term
Step 3: Plug everything into an = a1 × r(n-1)
Step 4: Simplify. Use a calculator if needed.
Common mistakes to avoid:
- Forgetting to subtract 1 from n in the exponent
- Confusing arithmetic and geometric sequences
- Using the wrong sign for a negative common ratio
Quick Reference
| What You Know | Formula to Use |
|---|---|
| Find nth term | an = a1 × r(n-1) |
| Find common ratio | r = an ÷ a(n-1) |
| Sum of n terms (r ≠ 1) | Sn = a1 × (1 - rn) ÷ (1 - r) |
| Sum to infinity (|r| < 1) | S∞ = a1 ÷ (1 - r) |
Bookmark this page. The formulas are straightforward once you stop overcomplicating them.