CalcLibrary

Arithmetic & Geometric Sequence Calculator

Find the nth term and the finite sum of an arithmetic or geometric sequence without mixing up the formulas. Choose the pattern first, enter a1, d or r, and n, then confirm the result with a first-terms preview.

Sequence rule preview
Examples

Check the 10th term and the sum of the first 10 terms.

n-th term a_n
31
Sum of the first n terms S_n
175
First terms preview
4, 7, 10, 13, 16, 19 ...

Arithmetic sequence: start at 4 and add 3 each time. Term 10 is 31, and the first 10 terms sum to 175.

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Examples

How It Works

Formula

an=a1+(n−1)da_n = a_1 + (n - 1)d

Sn=n2(2a1+(n−1)d)S_n = \frac{n}{2}\left(2a_1 + (n - 1)d\right)

an=a1⋅rn−1a_n = a_1 \cdot r^{n - 1}

Sn=a1(1−rn)1−r(r≠1)S_n = \frac{a_1(1 - r^n)}{1 - r} \quad (r \ne 1)

Sn=n⋅a1(r=1)S_n = n \cdot a_1 \quad (r = 1)

Variables

a1a_1

First term of the sequence

dd

Common difference in an arithmetic sequence

rr

Common ratio in a geometric sequence

nn

Number of terms, counted from 1

ana_n

n-th term

SnS_n

Sum of the first n terms

Choose arithmetic when the sequence changes by the same difference each step, and geometric when it changes by the same ratio. Then enter the first term, the relevant pattern value, and the positive whole-number term count n. The calculator returns the n-th term, the finite sum of the first n terms, and a short first-terms preview so you can confirm the pattern before trusting the answer.

Frequently Asked Questions

01How do I know whether a sequence is arithmetic or geometric?
Arithmetic sequences add or subtract the same amount each step, so they use a common difference d. Geometric sequences multiply by the same factor each step, so they use a common ratio r.
02How do I find the 15th term quickly?
Enter a1, the common difference or ratio, and n = 15. The calculator applies the standard nth-term formula directly, so you do not have to list all 15 terms by hand.
03Why does the geometric sum change when r = 1?
When r = 1 every term stays equal to a1, so the finite sum is just n copies of the same number. That is why the calculator switches to Sn = n · a1 for that case.
04Does this calculator handle infinite series or detect other patterns?
No. This tool is only for finite arithmetic and geometric sequences with values you already know. It does not test convergence, work with infinite-series rules, or identify arbitrary patterns automatically.

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