Sum of a Geometric Series Calculator
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📐 Sum of a Geometric Series Calculator – Instantly Find the Sum of Geometric Progressions
A Sum of a Geometric Series Calculator helps you quickly calculate the sum of the first n terms of a geometric series or even the infinite sum, if applicable.
Whether you’re a student, teacher, or data analyst, this tool saves time and reduces calculation errors.
🔍 What Is a Geometric Series?
A geometric series is a sequence where each term is multiplied by a common ratio (r) to get the next one.
Example series: 2+6+18+54+…2 + 6 + 18 + 54 + \dots2+6+18+54+…
Here, each term is multiplied by 3 → r = 3.
🧮 Geometric Series Sum Formulas
✅ Finite Geometric Series:
For first n terms: Sn=a⋅1−rn1−r(r≠1)S_n = a \cdot \frac{1 – r^n}{1 – r} \quad (r \neq 1)Sn=a⋅1−r1−rn(r=1)
Where:
- SnS_nSn = sum of the first n terms
- aaa = first term
- rrr = common ratio
- nnn = number of terms
🧠 Features of a Good Calculator
- Accepts decimals, negatives, fractions
- Solves both finite and infinite sums
- Provides step-by-step solutions
- Graphs geometric growth/decay (optional)
📚 Use Cases
- Mathematics: Sequences, series, induction proofs
- Finance: Compound interest, annuities
- Physics: Wave interference, signal processing
- Computer Science: Algorithm analysis (e.g., divide-and-conquer)
❓FAQs – Sum of a Geometric Series Calculator
🔹 What happens when r = 1?
All terms are the same, so the sum = a⋅na \cdot na⋅n
🔹 When is the infinite sum formula valid?
Only when ∣r∣<1|r| < 1∣r∣<1. Otherwise, the series diverges (no finite sum).
🔹 Can I input negative or fractional ratios?
✅ Yes — the calculator can handle any real number for r.
🔹 Can I use this for real-world problems like savings or investments?
Absolutely! It models compound growth, making it useful in finance and economics.