A geometric sequence multiplies by the same number at every step: 3, 6, 12, 24, and so on. Once you know the first term and that multiplier, called the common ratio, you can jump to any term without writing the whole list.
How it works
The nth term is a_n = a_1 x r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. The common ratio is any term divided by the previous term. The exponent is n – 1 because the first term needs zero multiplications to reach. A ratio above 1 grows, a ratio between -1 and 1 shrinks toward zero, and a negative ratio alternates sign at every step.
Step by step
- Write down the first term a_1.
- Find r by dividing the second term by the first, and confirm with another pair.
- Write down n, the term number you want.
- Use the formula: a_n = a_1 x r^(n-1).
- Compute the power first, then multiply by a_1.
- Check small cases by hand: the formula for n = 2 should return the second term.
Worked example
Sequence 5, 20, 80, 320. First term 5, ratio 4. The 6th term = 5 x 4^5 = 5 x 1024 = 5120. A shrinking example: 81, 27, 9, 3. Ratio = 27 / 81 = 1/3. The 5th term = 81 x (1/3)^4 = 81 / 81 = 1. To recover r from two terms: if a_1 = 3 and a_4 = 24, then r^3 = 24 / 3 = 8, so r = 2.
Common mistakes
- Using r^n instead of r^(n-1), which multiplies one extra time.
- Finding r from only one pair when the sequence is not geometric.
- Losing the sign of a negative ratio, which flips every other term.
- Rounding r early when it is a fraction, which compounds the error in later terms.
Try it with the free tool
You can follow every step above by hand, but the free tool below does the same calculation instantly. Try your own numbers in it, then check one result by hand so you trust the method as well as the answer. Open the full tool here: open the tool.
Checklist before you finish
- Write down every input number first, with its unit, before you calculate anything.
- Work through the steps in order, and do not skip the unit conversions.
- Check the answer against the worked example above: is it the right size, and does the direction make sense?
- Round only at the end, and keep the unrounded figure for any follow-up calculation.
Practice with your own numbers
Pick three cases of your own and work each one by hand. First, try a simple round-number case you can check mentally. Second, try a realistic case with awkward numbers, the kind you will actually meet. Third, try an edge case, such as a very small or very large input, and confirm the answer still behaves sensibly. If any case gives a surprising result, re-check the units and the formula step by step before you trust it.
If your answer looks wrong
Most errors come from a small set of causes. Check that you used the same units throughout, that you divided where the formula divides and multiplied where it multiplies, and that you applied any percentage as a decimal. Check the order of operations: powers and brackets come before multiplication and division, which come before addition and subtraction. Finally, test the method on the worked example in this guide. If your method reproduces the example answer, the method is right and the inputs deserve a second look.
Quick tips
If you are given a later term and the ratio, divide backwards to reach the first term, then use the formula. For very large n, the power grows fast, so check whether the answer size is plausible before trusting it.