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Geometric Series

Geometric Series

A series such as 2 + 6 + 18 + 54 + 162 or 3 + 1 + 1/3 + 1/9 + 1/27 + 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.

 

Formula: Sum = a₁(rⁿ-1)/(r-1) or a₁(1-rⁿ)/(1-r). Example: 3+1+1/3+1/9+1/27+1/81, where a₁=3, r=1/3, n=6; Sum

 

 

See also

Geometric sequence, infinite geometric series, arithmetic series

Key Formula

Sn=a1⋅1−rn1−r,r≠1S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1
Where:
  • SnS_n = The sum of the first n terms of the geometric series
  • a1a_1 = The first term of the series
  • rr = The common ratio (each term divided by the previous term)
  • nn = The number of terms being summed

Worked Example

Problem: Find the sum of the geometric series: 3 + 6 + 12 + 24 + 48 + 96.
Step 1: Identify the first term, common ratio, and number of terms. The first term is 3, each term is multiplied by 2, and there are 6 terms.
a1=3,r=63=2,n=6a_1 = 3, \quad r = \frac{6}{3} = 2, \quad n = 6
Step 2: Write out the sum formula for a finite geometric series.
Sn=a1⋅1−rn1−rS_n = a_1 \cdot \frac{1 - r^n}{1 - r}
Step 3: Substitute the values into the formula.
S6=3⋅1−261−2=3⋅1−64−1S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2} = 3 \cdot \frac{1 - 64}{-1}
Step 4: Simplify the fraction and multiply.
S6=3⋅−63−1=3⋅63=189S_6 = 3 \cdot \frac{-63}{-1} = 3 \cdot 63 = 189
Answer: The sum of the series is 189.

Another Example

This example uses a fractional common ratio (r = 1/2), producing a series with decreasing terms—a contrast to the first example where r > 1 and terms grow.

Problem: Find the sum of the first 5 terms of the geometric series where the first term is 200 and the common ratio is 1/2.
Step 1: Identify the given values. Here the common ratio is a fraction less than 1, so the terms decrease.
a1=200,r=12,n=5a_1 = 200, \quad r = \tfrac{1}{2}, \quad n = 5
Step 2: Compute r raised to the power n.
r5=(12)5=132r^5 = \left(\tfrac{1}{2}\right)^5 = \frac{1}{32}
Step 3: Substitute into the sum formula.
S5=200⋅1−1321−12=200⋅313212S_5 = 200 \cdot \frac{1 - \frac{1}{32}}{1 - \frac{1}{2}} = 200 \cdot \frac{\frac{31}{32}}{\frac{1}{2}}
Step 4: Simplify by dividing the fractions, then multiply by 200.
S5=200⋅3132⋅2=200⋅3116=620016=387.5S_5 = 200 \cdot \frac{31}{32} \cdot 2 = 200 \cdot \frac{31}{16} = \frac{6200}{16} = 387.5
Answer: The sum of the first 5 terms is 387.5.

Frequently Asked Questions

What is the difference between a geometric sequence and a geometric series?
A geometric sequence is an ordered list of numbers where each term is found by multiplying the previous term by a constant ratio (e.g., 2, 6, 18, 54). A geometric series is the sum of those terms (e.g., 2 + 6 + 18 + 54 = 80). The sequence lists the terms; the series adds them up.
When does a geometric series converge (have a finite sum)?
An infinite geometric series converges only when the absolute value of the common ratio is less than 1, that is, ∣r∣<1|r| < 1. In that case, the infinite sum equals S=a11−rS = \frac{a_1}{1 - r}. If ∣r∣≥1|r| \geq 1, the terms do not shrink toward zero, so the infinite series diverges and has no finite sum.
How do you find the common ratio of a geometric series?
Divide any term by the term immediately before it. For instance, in the series 5 + 15 + 45 + 135, the common ratio is r=15÷5=3r = 15 \div 5 = 3. This ratio must be the same between every pair of consecutive terms; if it varies, the series is not geometric.

Geometric Series vs. Arithmetic Series

Geometric SeriesArithmetic Series
Pattern between termsEach term is multiplied by a constant ratio rEach term increases by a constant difference d
Sum formula (finite)S_n = a₁ · (1 − rⁿ) / (1 − r)S_n = n/2 · (a₁ + aₙ)
Growth behaviorExponential growth or decay depending on rLinear growth or decline
Infinite sum possible?Yes, when |r| < 1No (diverges unless d = 0)
Example2 + 6 + 18 + 542 + 5 + 8 + 11

Why It Matters

Geometric series appear throughout algebra, precalculus, and calculus whenever quantities grow or shrink by a fixed percentage—compound interest, population models, and radioactive decay all follow geometric patterns. In calculus, understanding convergence of infinite geometric series is foundational to working with power series and Taylor series. Standardized tests like the SAT, ACT, and AP exams regularly include problems that require the finite or infinite geometric sum formula.

Common Mistakes

Mistake: Using the formula with r = 1, which causes division by zero.
Correction: When r = 1, every term equals a₁, so the sum is simply S_n = n · a₁. The standard formula requires r ≠ 1.
Mistake: Confusing the number of terms n with the last term's index when the series doesn't start at index 1.
Correction: Count how many terms are actually being summed. For example, the series from the 3rd term to the 7th term has n = 5 terms, not 7. Miscounting n gives an incorrect power of r in the formula.

Related Terms