Geometric Sequence Calculator

A geometric sequence multiplies by the same number each time, such as 2, 6, 18, 54. That number is the common ratio. This calculator finds any term, the sum of the first n terms and, when the ratio is between −1 and 1, the sum to infinity, with a chart showing the partial sums.

Try:

How to use the geometric sequence calculator

  1. Enter the first term a₁.
  2. Enter the common ratio r and the term number n.
  3. Press Calculate to see aₙ, Sₙ and the sum to infinity.

Formula

an=a1rn−1a_n = a_1 r^{n-1}
Sn=a1(1−rn)1−rS_n = \frac{a_1(1 - r^n)}{1 - r}
S∞=a11−r,  ∣r∣<1S_\infty = \frac{a_1}{1 - r},\; |r| < 1

Worked example: 2, 6, 18, … (n = 8)

  • First term a₁: 2
  • Common ratio r: 3
  • Term number n: 8

✓ Answer checked

a8=4374,S8=6560a_{8} = 4374,\quad S_{8} = 6560
First terms
2, 6, 18, 54, 162, 486, 1458, 4374
Sum to infinity
does not exist (∣r∣≥1)\text{does not exist } (|r| \ge 1)
01000200030004000500060001: 212: 823: 2634: 8045: 24256: 72867: 218678: 65608
Partial sums Sₙ for the first terms.
Step-by-step working (2 steps)

  1. nth term formula aₙ = a₁rⁿ⁻¹

    Start at a₁ and multiply by r, (n − 1) times.

    an=a1rn−1=2⋅(3)7=4374a_n = a_1 r^{n-1} = 2 \cdot (3)^{7} = 4374

  2. Sum formula Sₙ = a₁(1 − rⁿ)/(1 − r)

    Use the geometric series formula.

    Sn=a1(1−rn)1−r=6560S_n = \frac{a_1(1 - r^n)}{1 - r} = 6560
Formulas used
an=a1rn−1a_n = a_1 r^{n-1}
Sn=a1(1−rn)1−rS_n = \frac{a_1(1-r^n)}{1-r}
S∞=a11−r,  ∣r∣<1S_\infty = \frac{a_1}{1-r},\; |r|<1
How this was checked
  • ✓ Generating all 8 terms one by one gives the same last term and sum.

Frequently asked questions

When does a geometric series have a sum to infinity?

Only when −1 < r < 1. Then the terms shrink towards zero and the total settles at a₁ ÷ (1 − r).

What is 1 + 1/2 + 1/4 + … ?

It adds up to 2, because a₁ = 1 and r = ½, so the sum to infinity is 1 ÷ (1 − ½) = 2.

What is the common ratio?

The number you multiply by to get from one term to the next. Divide any term by the one before it to find it.

Related calculators

Arithmetic Sequence Calculator

Find the nth term and the sum of an arithmetic sequence from the first term and common difference.

Exponents & Powers Calculator

Raise any number to a power, including negative and fractional exponents.

Compound Interest Calculator

See how money grows with compound interest: yearly, monthly, daily or continuous compounding.

Related guides