Geometric Sequence Calculator
Any term and partial sum of a geometric sequence from first term and ratio.
What it does
A geometric sequence grows by multiplication where an arithmetic one grows by addition. Doubling each step gives 3, 6, 12, 24, while tripling gives 5, 15, 45, 135. The tool above jumps straight to any term and adds the run-up beside it.
The default starts at 3 with ratio 2 and asks for term 10. Repeated doubling reaches a10 = 1536, and the first ten terms add to S10 = 3069. One formula handles both: multiply the first term by the ratio raised to n minus 1.
Ratios below 1 shrink instead of grow. With ratio 0.5, the terms 2, 1, 0.5, 0.25 fade toward zero while the running sum approaches a ceiling it never crosses. Negative ratios swing signs each step, which models alternating patterns in signals and finance.
Sums matter wherever repeated growth stacks: compound balances, spreading processes, and shrinking doses. When the ratio equals exactly 1, every term repeats and the sum is just the first term times the count.
How to use
- Type the first term a1.
- Type the common ratio r, the multiplier between terms.
- Type which term number n you want.
- Read the term first, then the sum of terms 1 through n.
- Copy the explicit rule to generate further terms by hand.
Formula
- a1
- First term of the sequence.
- r
- Common ratio between consecutive terms.
- n
- Term position, 1 to 1,000,000.
The sum form needs r different from 1; at r = 1 the sequence is flat and the sum is plain multiplication. Huge ratios overflow fast, so the tool stops instead of printing infinity.
Worked example
Example: first term 3, ratio 2, term 10
Defaults: a1 3, r 2, n 10.
- Powers: 2^9 = 512 and 2^10 = 1024.
- Term: a10 = 3 x 512 = 1536.
- Sum top: 3 x (1024 minus 1) = 3069.
- Divide by (2 minus 1) = 1, so S10 = 3069.
- Tool shows a10 = 1536, S10 = 3069, rule an = 3 x 2^(n-1).
Reference table
| n | Term | Sum so far |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 6 | 9 |
| 3 | 12 | 21 |
| 4 | 24 | 45 |
| 5 | 48 | 93 |
Limits
- Finite prefixes only; infinite sums need convergence checks.
- Rounding shows 4 decimals, so tiny fractional tails clip.
- Overflow guard stops ratios that explode past finite floats.
- Models exact math, not noisy measured growth.
Common mistakes
- Calling geometric doubling arithmetic.
- Using r^n instead of r^(n-1) for the term.
- Dividing by (r minus 1) when r equals 1.
- Mixing the term index starting at 0 versus 1.
- Reading the sum as the last term instead of the total.
Try it yourself
Quick checks. Answers sit tucked under each question.
First term 5, ratio 3, term 4?
a4 = 5 x 27 = 135. S4 = 5 x 80 / 2 = 200.
First term 7, ratio 1, first 5 sum?
Every term is 7, so a5 = 7 and S5 = 35.
First term 2, ratio 0.5, term 4?
a4 = 2 x 0.125 = 0.25. S4 = 3.75.
Questions
How do I spot a geometric sequence?
Divide each term by the last. A constant quotient is the ratio.
Arithmetic versus geometric?
Arithmetic adds a fixed difference; geometric multiplies by a fixed ratio.
What if the ratio is 1?
All terms equal a1, and the sum of n terms is a1 times n.
What if the ratio is negative?
Signs alternate each term, with size growing or shrinking by |r|.
Why n minus 1 in the exponent?
The first term uses r^0 = 1, so term n applies the multiplier n minus 1 times.
Can the sum exceed the last term?
Yes for positive ratios above 1, since it stacks every term so far.