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.