Geometric sequence calculator
Find the nth term and partial sum of a geometric sequence.
What this calculator covers
Use this geometric sequence calculator to evaluate a sequence that scales by the same ratio at each step.
The walkthrough keeps the power rule for the nth term and the geometric-series sum formula visible, including the special case where the ratio is 1.
Frequently asked questions
- What makes a sequence geometric?
- A sequence is geometric when each term is found by multiplying the previous term by the same fixed number, called the common ratio. Doubling (ratio = 2) and halving (ratio = 0.5) are familiar examples.
- What happens when the common ratio equals 1?
- When the ratio is exactly 1, every term equals the first term and the partial sum is simply the first term multiplied by the number of terms — the standard formula does not apply because its denominator becomes zero.
- What does the partial sum represent?
- The partial sum is the total you get by adding all the terms from the first through the nth term. It answers "how much accumulates over n steps?" rather than "what is the value at step n?"
- Can the common ratio be negative or a fraction?
- Yes. A negative ratio causes the terms to alternate in sign, and a ratio between −1 and 1 (excluding 0) causes the terms to shrink toward zero. Both cases are handled by the same nth-term formula.
Tool
Run the calculation
Result
RESULT · SEQUENCE
â„–214
Primary result
a5 = 48; S5 = 93
Starting from 3 and multiplying by 2 each step gives a5 = 48 and S5 = 93.
- Preview terms
- 3, 6, 12, 24, 48
- Nth term
- 48
- Partial sum
- 93
- Term index
- 5
Step-by-step solution
- 1.Use a_n = a1 · r^(n - 1) = 3 · 2^(5 - 1) to get 48.
- 2.Use the geometric-series sum formula S_n = a1(1 - r^n) / (1 - r).
- 3.Read the partial sum S5 as 93.
Walkthrough
Visual walkthrough
Geometric sequences scale by a constant ratio, so powers of the ratio control both the nth term and the running total.
01
Fix the growth ratio
r = 2
Every new term is the previous term multiplied by the same ratio.
02
Jump straight to the nth term
a5 = 48
Exponentiation counts how many ratio jumps happen between the first term and the requested position.
03
Add the geometric run
The sum formula captures the total effect of all the ratio-scaled terms from the first through the nth term.
S5 = 93