Direct answer
An arithmetic sequence changes by a constant difference, while a geometric sequence changes by a constant ratio; inspect consecutive terms with the operation appropriate to the data.
What this calculation tells you
Arithmetic sequences have equally spaced values and linear growth in the term number. Geometric sequences repeatedly multiply and therefore model exponential growth, decay, or sign alternation.
Finite data can match several possible rules, so context and a stated model matter in addition to the first few terms.
Where it is used
Finance education
Contrast constant cash changes with constant percentage changes without replacing a full financial model.
Science
Represent idealised repeated increments or proportional changes.
Operations
Plan regular step schedules or multiplicative scaling under stable assumptions.
Computing
Analyse linear versus multiplicative resource patterns.
When this guide helps
- Finding a missing term.
- Computing a finite series sum.
- Comparing linear and percentage growth.
- Testing whether observed data actually follows a proposed sequence.
Test differences and ratios
Subtract neighbouring terms for arithmetic structure. Divide neighbouring terms for geometric structure only when the divisor is nonzero and the ratio is meaningful.
Separate terms from sums
A term formula locates one position; a series formula accumulates several terms. Confusing the two produces answers with different meanings.
Treat extrapolation cautiously
Real processes may change regimes, round values, or include noise. A simple sequence describes its assumptions rather than proving the future follows them.
Common mistakes
Typical mistakes include using a ratio across a zero term and assuming any smooth-looking list is geometric.
- Index terms consistently.
- Check units of differences and ratios.
- State whether the sequence is finite.
Worked case: arithmetic sequence
A sequence starts at 5 and adds 3 each term.
The tenth term is 5+9x3=32. Sum of ten terms is 10/2 x (5+32)=185.
a10=32 and S10=185.
A constant difference, not a constant percentage, defines the model.
Reproduce this worked caseOpen Arithmetic Sequence Calculator
Worked case: geometric sequence
A sequence starts at 2 and multiplies by 3.
The sixth term is 2x3⁵=486. Sum of six terms is 2(3⁶-1)/(3-1)=728.
a6=486 and S6=728.
A constant ratio creates multiplicative growth and a very different total.
Reproduce this worked caseOpen Arithmetic Sequence Calculator
Compare arithmetic and geometric sequences cases before generalising
Check consecutive differences and ratios before choosing the model. A short noisy dataset may fit neither exact sequence and should not be forced into one.
| Sequence | Invariant | Selected term | Sum |
|---|---|---|---|
| 5, 8, 11, ... | Difference 3 | a10=32 | S10=185 |
| 2, 6, 18, ... | Ratio 3 | a6=486 | S6=728 |
arithmetic and geometric sequences: calculation checklist
- Identify term numbering
- Test difference or ratio
- Use n-1 exponent/steps
- Handle r=1 separately in sum formula
- Verify initial terms
Practical questions
Frequently asked questions
Can a sequence be both arithmetic and geometric?
A constant sequence can satisfy both under appropriate definitions; otherwise the structures generally differ.
What if the ratio is negative?
A geometric sequence then alternates signs while magnitudes scale by the ratio's absolute value.
Does a geometric sequence always grow?
No. Magnitudes decay when the absolute ratio is below one and remain constant when it equals one.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.