Case 1
- First term a₁
- 2
- Common ratio r
- 3
- Term position n
- 5
162.0000
Open with these valuesGeometric Sequence Calculator
162.0000
Result: 162.0000Every term is the previous one multiplied by a fixed ratio r, so the nth term is a × r^(n − 1). Counting starts at n = 1, and the first term needs no multiplication — that is where the minus one comes from. With a = 2 and r = 3 the fifth term is 2 × 81 = 162.
Worked examples
Case 2
- First term a₁
- 100
- Common ratio r
- 0.5
- Term position n
- 3
25.0000
Open with these valuesCase 3
- First term a₁
- 5
- Common ratio r
- -2
- Term position n
- 4
-40.0000
Open with these valuesHow it's calculated
aₙ = a × r^(n − 1)
- StepEnter the first term — the value the sequence starts at.
- StepEnter the common ratio: the fixed factor each step multiplies by.
- StepEnter the position you want; n = 1 is the first term itself.
- ResultRead the nth term. The table shows the running sum beside it.
Reference table
| a, r, n | Sum of the first n terms | nth term |
|---|---|---|
| 3, 1, 4 | 12 | 3 |
| 100, 0.5, 3 | 175 | 25 |
| 5, -2, 4 | -25 | -40 |
| 2, 3, 5 | 242 | 162 |
| 1, 2, 10 | 1023 | 512 |
Questions
What is a geometric sequence?
A geometric sequence is a list of numbers where each term comes from multiplying the previous one by a fixed factor — the common ratio r: a, ar, ar², … For example 2, 6, 18, 54, … has first term 2 and common ratio 3. The ratio between any two neighbours is always the same.
How do I find the nth term?
Use aₙ = a × r^(n − 1), where a is the first term, r is the common ratio and n is the position you want. For 2, 6, 18, … the fifth term is 2 × 3⁴ = 162. The exponent is n − 1 because the first term needs zero multiplications by r.
What does the common ratio tell me?
Its size sets the speed: a ratio above 1 in absolute value grows the terms fast (2, 6, 18, …), below 1 shrinks them toward zero (100, 50, 25, …) and exactly 1 keeps every term equal to the first. A negative ratio flips the sign at every step (5, −10, 20, −40, …). This is the same machinery behind compound interest and population models.
How do I get the sum of the first n terms?
The sum is a × (1 − rⁿ) ÷ (1 − r) whenever r is not 1, and simply a × n when it is. For a = 2, r = 3, n = 5 that gives 2 × (1 − 243) ÷ (1 − 3) = 242. The middle column of the table above carries this sum for every row.
How is a geometric sequence different from an arithmetic one?
A geometric sequence multiplies by a constant ratio each step, so the ratios are equal (2, 6, 18, 54). An arithmetic sequence adds a constant difference each step, so the differences are equal (2, 5, 8, 11). Use this calculator for the multiplicative kind.
Sources and last check
Information, not professional advice.