How this calculator works
- Step 1Arithmetic+ dAdd the same difference each time.
- Step 2Geometric× rMultiply by the same ratio each time.
- Step 3Closed formsaₙ and SₙJump straight to any term or sum.
You do not need to list every term. A closed formula gives the nth term and the sum of the first n terms in one step.
Worked example: 3, 7, 11, …
- Term
aₙ = a + (n − 1)d- Sum
Sₙ = n(2a + (n − 1)d) ÷ 2
- aFirst term 3
- dDifference 4
- nTerms 20
- 20th term3 + 19 × 479
- Sum of 2020 × (6 + 76) ÷ 2820
The same formula gives the famous school sum 1 + 2 + … + 100 = 5,050.
Growing or shrinking
- 2, 6, 18, …39,36610th term with ratio 3; the ten terms total 59,048.
- 81, 40.5, …2.531256th term with ratio 0.5; the six total 159.46875.
A ratio above 1 grows fast; a ratio between 0 and 1 shrinks towards zero.
Assumptions and limits
- Any startNegative and decimal terms work.
- Ratio 1Handled without dividing by zero.
- Huge numbersResults past 10³⁰⁰ are reported, not shown.
- Ten listedThe first ten terms are shown.
Related number tools
Sequences grow by a fixed step or a fixed ratio. Geometric growth is the same maths as compound interest.
A geometric sequence with ratio 1.1 is money growing at 10% a year; the compound interest calculator applies that to deposits. To raise a number to any power, use the exponent calculator.
Common mistakes
Mistake: Using n instead of n − 1 for the nth term.
Instead: The first term already counts once.
Mistake: Treating 2, 4, 8 as arithmetic.
Instead: Check: do you add or multiply?
Mistake: Dividing by r − 1 when r = 1.
Instead: The sum is simply n × a.
Checking the method
A separate script builds each term in exact fractions and adds them in a loop, without the closed formulas.
Read how we test our calculators, or browse more maths calculators.