Geometric sequence term is one of 4 sequences formulas in the algebra section of this library, and it is used at high school level.
Why geometric sequence term works
Each move to the next term multiplies by r, and reaching term n from term 1 takes n minus 1 of those multiplications. That is why the exponent lags the term number by one: term 1 has been multiplied zero times, and r to the zero is 1.
What each symbol means
$a_1$ is the first term, $r$ the common ratio, and $n$ the term number.
Geometric sequence term: when it holds
$n$ is a positive integer; when dividing to find $r$, the earlier term must be nonzero.
When it stops applying
If any term is zero the ratio cannot be recovered by dividing, and multiplying can never escape zero afterwards. A list like 5, 0, 0, 0 is not geometric in any useful sense, and no nonzero r reproduces it.
Geometric sequence term: a worked example
If $a_1=3$ and $r=2$, then $a_6=3(2^5)=96$.
The mistake to avoid
What people do: Writing a1 times r to the n, giving 192 for a sequence that starts at 3 and doubles.
Why it goes wrong: The sixth term is 3 times 2 to the fifth, which is 96. Using n as the exponent multiplies one extra time, so every answer comes out r times too large.
Do this instead: Check the formula on term 1 before you use it: the exponent must collapse to zero there so the answer is just a1.
Geometric sequence term: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a_1$ is the first term, $r$ the common ratio, and $n$ the term number.
- Check the conditions before substituting. $n$ is a positive integer; when dividing to find $r$, the earlier term must be nonzero.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Sequences
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where geometric sequence term comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about geometric sequence term
How do I find r from two terms that are not adjacent?
Divide the later value by the earlier one and take the root matching the gap. From a second term of 6 and a fifth term of 48, the ratio cubed is 8, so r is 2.
Can the common ratio be negative?
Yes, and the signs then alternate. Starting at 5 with r = -2 produces 5, -10, 20, -40, where the size grows steadily while the sign flips every step.
What happens when the ratio is between -1 and 1?
The terms shrink toward zero without ever reaching it. This is the behaviour behind half-lives and depreciation, where each period keeps a fixed fraction of the previous amount.
Is compound interest a geometric sequence?
Yes, with r equal to 1 plus the rate. A 1000 balance at 5 percent per year gives 1050, 1102.50, then 1157.625, which is exactly 1000 times 1.05 cubed.