sum of geometric series calculator

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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