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Geometric Sequence — Definition, Formula & Examples

Geometric Sequence
Geometric Progression

A sequence such as 2, 6, 18, 54, 162 or 3,1,13,19,127,181\displaystyle 3, 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81} which has a constant ratio between terms. The first term is a1, the common ratio is r, and the number of terms is n.

Explicit Formula: an=a1rn−1a_n = a_1 r^{n-1}

Example 1: 2, 6, 18, 54, 162 has a1=2,a_1 = 2\text{,} r=3,r = 3\text{,} and n=5.n = 5\text{.}
The explicit formula is an=2⋅3n−1.a_n = 2 \cdot 3^{n-1}\text{.}

Example 2: 3,1,13,19,127,181\displaystyle 3, 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81} has a1=3,a_1 = 3\text{,} r=13,\displaystyle r = \frac{1}{3}\text{,} and n=6.n = 6\text{.}
The explicit formula is an=3(13)n−1.\displaystyle a_n = 3\left(\frac{1}{3}\right)^{n-1}\text{.}

See also

Geometric series, infinite geometric series, arithmetic sequence

Key Formula

an=a1⋅r n−1a_n = a_1 \cdot r^{\,n-1}
Where:
  • ana_n = The nth term of the sequence (the term you want to find)
  • a1a_1 = The first term of the sequence
  • rr = The common ratio (the constant multiplier between consecutive terms)
  • nn = The position number of the term in the sequence

Worked Example

Problem: Find the 8th term of the geometric sequence 5, 15, 45, 135, …
Step 1: Identify the first term.
a1=5a_1 = 5
Step 2: Find the common ratio by dividing any term by the term before it.
r=155=3r = \frac{15}{5} = 3
Step 3: Write the general term formula and substitute the known values with n = 8.
a8=5⋅3 8−1=5⋅37a_8 = 5 \cdot 3^{\,8-1} = 5 \cdot 3^7
Step 4: Calculate the power of 3.
37=2,1873^7 = 2{,}187
Step 5: Multiply to get the final answer.
a8=5⋅2,187=10,935a_8 = 5 \cdot 2{,}187 = 10{,}935
Answer: The 8th term is 10,935.

Another Example

This example uses a fractional common ratio (r = 1/2), showing that geometric sequences can decrease. The terms shrink toward zero rather than growing, which is a common variation students encounter.

Problem: A geometric sequence has a first term of 256 and a common ratio of 1/2. Find the 6th term.
Step 1: Write down the known values.
a1=256,r=12,n=6a_1 = 256, \quad r = \tfrac{1}{2}, \quad n = 6
Step 2: Substitute into the nth-term formula.
a6=256⋅(12)6−1=256⋅(12)5a_6 = 256 \cdot \left(\tfrac{1}{2}\right)^{6-1} = 256 \cdot \left(\tfrac{1}{2}\right)^5
Step 3: Evaluate the power.
(12)5=132\left(\tfrac{1}{2}\right)^5 = \frac{1}{32}
Step 4: Multiply to find the 6th term.
a6=256⋅132=8a_6 = 256 \cdot \frac{1}{32} = 8
Answer: The 6th term is 8.

Frequently Asked Questions

How do you find the common ratio of a geometric sequence?
Divide any term by the term immediately before it: r=an+1anr = \frac{a_{n+1}}{a_n}. If the sequence is truly geometric, this ratio is the same for every pair of consecutive terms. For example, in 4, 12, 36, the common ratio is 12÷4=312 \div 4 = 3.
What is the difference between a geometric sequence and a geometric series?
A geometric sequence is the ordered list of terms (e.g., 2, 6, 18, 54). A geometric series is the sum of those terms (e.g., 2 + 6 + 18 + 54 = 80). The sequence names the individual values; the series adds them together.
Can the common ratio of a geometric sequence be negative?
Yes. When rr is negative, the terms alternate in sign. For instance, with a1=3a_1 = 3 and r=−2r = -2, the sequence is 3, −6, 12, −24, … . The formula an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} still applies; just be careful with the sign when raising a negative number to a power.

Geometric Sequence vs. Arithmetic Sequence

Geometric SequenceArithmetic Sequence
Pattern between termsEach term is multiplied by a constant ratio rEach term has a constant difference d added
nth-term formulaan=a1⋅rn−1a_n = a_1 \cdot r^{n-1}an=a1+(n−1)da_n = a_1 + (n-1)d
Growth behaviorExponential (grows or decays rapidly)Linear (grows or decreases steadily)
Example2, 6, 18, 54, 1622, 5, 8, 11, 14
Key operationMultiplication (or division)Addition (or subtraction)

Why It Matters

Geometric sequences model real-world situations involving repeated multiplication, such as compound interest, population growth, radioactive decay, and the behavior of bouncing balls. They appear throughout algebra, pre-calculus, and calculus—especially when you study series and convergence. Understanding them also builds the foundation for working with exponential functions, since the nth-term formula is itself an exponential expression.

Common Mistakes

Mistake: Using n instead of n − 1 as the exponent.
Correction: The exponent is n − 1 because the first term corresponds to n = 1, and r1−1=r0=1r^{1-1} = r^0 = 1, which correctly gives a1a_1. Writing an=a1⋅rna_n = a_1 \cdot r^n gives a value that is one position ahead of the intended term.
Mistake: Adding the common ratio instead of multiplying by it.
Correction: A geometric sequence is defined by multiplication, not addition. If you add a constant, you have an arithmetic sequence. Always check: divide a term by the previous one to confirm the constant ratio.

Related Terms