Geometric Sequence Calculator
This calculator helps you work with geometric sequences (geometric progressions). You can find the nth term or the sum of the first n terms.
Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant ratio, r.
- nth term formula: aₙ = a₁ × r^(n-1)
- Sum of first n terms (r ≠ 1): Sₙ = a₁ × (1 - r^n) / (1 - r)
Example: Sequence: 2, 6, 18, 54,… → a₁ = 2, r = 3 → 5th term: 2 × 3^(5-1) = 162
How the Calculator Works
The calculator automatically:
Step 1: Takes input values: a₁, r, n
Step 2: Selects calculation type: nth term or sum
Step 3: Applies the geometric sequence formula
Step 4: Displays result instantly
Examples
1. a₁ = 3, r = 2, n = 5 → nth term: 3 × 2^(5-1) = 48
2. a₁ = 1, r = 3, n = 4 → sum of first 4 terms: 1 × (1-3^4)/(1-3) = (1-81)/(-2) = 40
3. a₁ = 5, r = 0.5, n = 6 → nth term: 5 × 0.5^(6-1) = 0.15625
Why Geometric Sequences Matter
Geometric sequences appear in finance (interest, investments), physics, and exponential growth or decay problems. They help identify patterns and calculate terms and sums quickly.
Common Mistakes to Avoid
- Using wrong formula (nth term vs sum)
- Forgetting that r ≠ 1 for the sum formula
- Miscalculating powers
Important: For sum when r = 1, use Sₙ = n × a₁, not the standard formula.
Correct: Check r value before applying the formula.
Frequently Asked Questions
Can r be negative?
Yes. Sequences can alternate signs if the ratio is negative.
Can I use decimals?
Yes. a₁, r, and n can be decimals or integers.
What if n = 1?
The first term a₁ is the nth term; sum = a₁.
Conclusion
This geometric sequence calculator is a fast and reliable tool to find the nth term or sum of the first n terms, helping students, professionals, and math enthusiasts solve geometric progression problems easily.