Sum of Finite Geometric Series
otals
A Sum of Geometric Series Calculator quickly computes the sum of a finite or infinite geometric series by using a standard formula. It’s especially useful in algebra, finance, computer science, and physics.
📘 What Is a Geometric Series?
A geometric series is a sum of terms where each term is found by multiplying the previous one by a fixed value called the common ratio rrr. General form: a+ar+ar2+ar3+⋯\text{General form: } a + ar + ar^2 + ar^3 + \cdotsGeneral form: a+ar+ar2+ar3+⋯
Where:
- aaa = first term
- rrr = common ratio
- nnn = number of terms (for finite series)
🔢 Geometric Series Formulas
✅ Finite Geometric Series:
Sn=a⋅1−rn1−r,r≠1S_n = a \cdot \frac{1 – r^n}{1 – r}, \quad r \ne 1Sn=a⋅1−r1−rn,r=1
✅ Infinite Geometric Series (converging):
S=a1−r,∣r∣<1S = \frac{a}{1 – r}, \quad |r| < 1S=1−ra,∣r∣<1
🧠 Example – Finite Series
Find the sum of the first 4 terms of:
2, 4, 8, 16
- a=2a = 2a=2, r=2r = 2r=2, n=4n = 4n=4
S4=2⋅1−241−2=2⋅1−16−1=2⋅−15−1=30S_4 = 2 \cdot \frac{1 – 2^4}{1 – 2} = 2 \cdot \frac{1 – 16}{-1} = 2 \cdot \frac{-15}{-1} = 30S4=2⋅1−21−24=2⋅−11−16=2⋅−1−15=30
🧠 Example – Infinite Series
Find the sum of:
5+2.5+1.25+…5 + 2.5 + 1.25 + \dots5+2.5+1.25+…
- a=5a = 5a=5, r=0.5r = 0.5r=0.5
S=51−0.5=50.5=10S = \frac{5}{1 – 0.5} = \frac{5}{0.5} = 10S=1−0.55=0.55=10
🧰 Features of a Good Geometric Series Calculator
- Input:
- First term (aaa)
- Common ratio (rrr)
- Number of terms (nnn) or choose infinite
- Output:
- Exact and decimal value of the sum
- Step-by-step calculation
- Optional:
- Graph of the series
- Term-by-term breakdown
❓ FAQs – Sum of Geometric Series Calculator
🔹 What is the difference between finite and infinite geometric series?
- Finite: Ends after nnn terms
- Infinite: Continues forever; only converges if ∣r∣<1|r| < 1∣r∣<1
🔹 Can the common ratio be negative?
Yes — it will cause terms to alternate (e.g., 5, -10, 20, -40…)
🔹 What happens if r=1r = 1r=1?
The formula becomes invalid. Instead, use: Sn=a⋅nS_n = a \cdot nSn=a⋅n
🔹 Can this calculator work for decimal or fractional ratios?
Absolutely. It supports all real number inputs for aaa, rrr, and nnn.
🔹 Is it useful in real life?
Yes — it’s used in:
- Compound interest
- Signal processing
- Population modeling
- Computer algorithms