Number Sequence Calculator
Find the nth term and the sum of an arithmetic or geometric sequence.
Formula and sources updated · Data sources · 2View algorithm, constants and default inputs (JSON)
About this calculator
An arithmetic sequence adds the same difference at every step, while a geometric sequence multiplies by the same ratio. Recognising which one you are looking at is the key to predicting the next terms.
Worked example
calculated with the default values below
First term 3 · Common difference or ratio 4 · Term number (n) 10 · Sequence type Arithmetic (add each step) Nth term 39.0000 · Sum of first n terms 210.0000 · Next term 43.0000 · Average of the series 21.0000
Method and assumptions
arith: a_k = a₁+(k−1)d, S = k(2a₁+(k−1)d)/2; geom: a_k = a₁r^(k−1), S = a₁(r^k−1)/(r−1) For geometric r = 1, S = a₁×k; next term = a_k+d (arithmetic) or a_k×r (geometric); mean = S/k; non-finite results are flagged as too large. An arithmetic sequence adds the same difference at every step, while a geometric sequence multiplies by the same ratio. Recognising which one you are looking at is the key to predicting the next terms. The references explain definitions and methods; any extra preset must be checked against the assumptions on this page.
Data sources
The references explain definitions and methods; any extra preset must be checked against the assumptions on this page.
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
arith: a_k = a₁+(k−1)d, S = k(2a₁+(k−1)d)/2; geom: a_k = a₁r^(k−1), S = a₁(r^k−1)/(r−1)
- a_k
- k-th term
- S
- sum of first k terms
- a₁
- first term
- d
- common difference
- r
- common ratio
- k
- number of terms
For geometric r = 1, S = a₁×k; next term = a_k+d (arithmetic) or a_k×r (geometric); mean = S/k; non-finite results are flagged as too large.
Frequently asked questions
Why does the geometric sum blow up faster than the arithmetic one?
In a geometric sequence each step multiplies by the ratio, so the 10th term of 3, 4, … is 3×4^9, a huge number, while an arithmetic sequence only adds the same difference and reaches 39 at term 10 with a sum of 210. The geometric sum formula divides by (ratio−1), which fails at ratio 1 and is why the tool special-cases that value.
Which inputs and limits apply to Number Sequence Calculator?
Use the units shown beside each field. These are form limits, not recommended settings. First term: -1,000,000,000–1,000,000,000; Common difference or ratio: -1,000,000–1,000,000; Term number (n): 1–10,000 arith: a_k = a₁+(k−1)d, S = k(2a₁+(k−1)d)/2; geom: a_k = a₁r^(k−1), S = a₁(r^k−1)/(r−1) For geometric r = 1, S = a₁×k; next term = a_k+d (arithmetic) or a_k×r (geometric); mean = S/k; non-finite results are flagged as too large. An arithmetic sequence adds the same difference at every step, while a geometric sequence multiplies by the same ratio. Recognising which one you are looking at is the key to predicting the next terms. The references explain definitions and methods; any extra preset must be checked against the assumptions on this page.
What is a complete worked example for First term = 3; Common difference or ratio = 4?
Inputs: First term = 3; Common difference or ratio = 4; Term number (n) = 10; Sequence type = Arithmetic (add each step). Calculation: arith: a_k = a₁+(k−1)d, S = k(2a₁+(k−1)d)/2; geom: a_k = a₁r^(k−1), S = a₁(r^k−1)/(r−1). Results: Nth term = 39.0000; Sum of first n terms = 210.0000; Next term = 43.0000; Average of the series = 21.0000.