Terms and gaps have different counts
From term 1 to term n there are n - 1 changes. Using n differences in an arithmetic rule produces the following term rather than the requested one.
Arithmetic and Geometric Sequence Calculator with Steps
Find a selected term and partial sum for arithmetic or geometric sequences with formula selection, substitution and pattern checks.
Free learning toolValidated inputs and visible working
YOUR RESULT
Your requested sequence term
Enter your valuesChoose an example or type the known values, then calculate to see the answer and working.
Terms, gaps and growth patterns
Difference and ratio identify families
A constant subtraction between neighbours indicates arithmetic structure. A constant division by the preceding non-zero term indicates geometric structure. A sequence can be neither.
A partial sum is not a term
The nth term names one position; the sum adds every term from the first through n. Questions about total savings or accumulated distance often require the sum, not only the last value.
Worked example: Find the 10th term of 3, 5, 7, ...
The difference is 2 and there are nine increases from the first term to the tenth.
Prediction first
Estimate the sign, scale, range or likely form of the answer before calculating. A prediction makes an input error easier to notice.
- Step
Identify an arithmetic sequence because each term increases by 2.
- Step
Use aₙ = a₁ + (n - 1)d with a₁ = 3, n = 10 and d = 2.
- Step
Calculate a₁₀ = 3 + 9(2) = 21.
- Step
Check neighbouring terms: the ninth is 19 and adding the common difference gives 21.
How a term rule locates a distant value
A sequence is an ordered list generated by a rule. Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. This calculator finds a requested term and the sum from the first term to that position. Selecting the sequence type is a mathematical decision, so inspect several consecutive terms before entering the difference or ratio.
Identify the sequence family before calculating
Use it when the first term, common difference or ratio, and positive term number are known. It supports standard arithmetic and geometric rules up to term 10,000. It does not infer a unique pattern from a short list because many rules can fit the same opening terms. Name the intended rule or derive it from the question first.
Sequence errors caused by the wrong position or rule
| Mistake | Why it fails | Better check |
|---|---|---|
| Using n instead of n - 1 | The first term already occupies position one and needs no increase. | Count the gaps between positions or write n - 1 explicitly. |
| Calling a changing difference arithmetic | If the differences vary, the arithmetic formula cannot represent every term. | Compare at least two consecutive differences before selecting the mode. |
| Mixing term and sum formulas | A large accumulated total can be mistaken for the nth term. | Write whether the question asks for one position or all positions up to it. |
Arithmetic and Geometric Sequence Calculator questions
Can the common ratio be negative?
Yes. A negative ratio creates alternating signs. Large term numbers can grow quickly, so interpret the magnitude carefully.
What happens when the geometric ratio is one?
Every term equals the first term and the finite sum is the first term multiplied by the number of terms.
Can this discover the rule from any list?
No. It evaluates a stated arithmetic or geometric model. Establish that model from consistent differences, ratios or problem information first.
Use this term to test the full pattern
The result points to related practice, but the best next step depends on what the learner could explain without help.