Geometric Sequence Sum Calculator
📐 Geometric Sequence Sum Calculator – Quickly Find the Sum of Any Geometric Sequence
A Geometric Sequence Sum Calculator is a helpful tool for calculating the sum of a finite or infinite geometric sequence based on a few inputs: the first term, common ratio, and number of terms.
This calculator is commonly used in math education, financial modeling, and scientific calculations where exponential growth or decay patterns occur.
📘 What Is a Geometric Sequence?
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant value, known as the common ratio (r).
General form:
a, ar, ar2, ar3, …a,\ ar,\ ar^2,\ ar^3,\ \ldotsa, ar, ar2, ar3, …
Where:
- aaa = first term
- rrr = common ratio
- nnn = number of terms
🔢 Formula to Calculate Geometric Sequence Sum
✅ Finite Geometric Sequence:
Sn=a⋅1−rn1−r,for r≠1S_n = a \cdot \frac{1 – r^n}{1 – r},\quad \text{for } r \ne 1Sn=a⋅1−r1−rn,for r=1
✅ Infinite Geometric Sequence (when ∣r∣<1|r| < 1∣r∣<1):
S=a1−rS = \frac{a}{1 – r}S=1−ra
🧠 Example – Finite Sequence
Given:
- First term a=3a = 3a=3
- Common ratio r=2r = 2r=2
- Number of terms n=4n = 4n=4
S4=3⋅1−241−2=3⋅1−16−1=3⋅−15−1=45S_4 = 3 \cdot \frac{1 – 2^4}{1 – 2} = 3 \cdot \frac{1 – 16}{-1} = 3 \cdot \frac{-15}{-1} = 45S4=3⋅1−21−24=3⋅−11−16=3⋅−1−15=45
🧰 Features of a Geometric Sequence Sum Calculator
- Input fields:
- First term aaa
- Common ratio rrr
- Number of terms nnn or “∞”
- Output:
- Step-by-step solution
- Final sum (exact and decimal form)
- Optional graph of terms
📊 Applications
- Compound interest
- Computer algorithms
- Population growth modeling
- Physics (e.g., radioactive decay)
- Signal processing
❓ FAQs – Geometric Sequence Sum Calculator
🔹 Can I calculate the sum of an infinite geometric sequence?
Yes, if the absolute value of the ratio is less than 1: ∣r∣<1|r| < 1∣r∣<1
🔹 What if the common ratio is 1?
The sum becomes a repeated value: Sn=a⋅nS_n = a \cdot nSn=a⋅n
🔹 Can the common ratio be negative?
Yes. The sequence will alternate between positive and negative terms.
🔹 Does the order matter?
No — geometric sequences are structured by multiplication, not order of terms.
🔹 Can it handle decimals or fractions?
Absolutely — most calculators accept decimal or fractional inputs.