Number Sequence Calculator
Work out the nth term and the sum of the first terms in an arithmetic or geometric sequence.
How to use the number sequence calculator
- Choose whether the sequence is arithmetic (fixed difference) or geometric (fixed ratio).
- Enter the starting value a₁, the difference or ratio, and how many terms n you want included.
- The calculator shows the nth term and the sum of the first n terms automatically.
How the sequence is calculated
An arithmetic sequence has a fixed number d that's added for each new term, for example 2, 5, 8, 11, … with d = 3. The formulas are:
Sum: Sₙ = (n ÷ 2) × (2a₁ + (n − 1) × d)
Worked example: an arithmetic sequence starting at 2 and increasing by 3 has as its 10th term 2 + (10 − 1) × 3 = 29. The sum of the first ten terms is (10 ÷ 2) × (2 × 2 + 9 × 3) = 5 × 31 = 155. The sum formula rests on a trick: add the sequence to itself in reverse and every pair comes out equal, so the total is simply the number of pairs times the pair value.
A geometric sequence has a fixed number r that each term is multiplied by, for example 2, 6, 18, 54, … with r = 3. The formulas are:
Sum: Sₙ = a₁ × (rn − 1) ÷ (r − 1)
Arithmetic or geometric — check the difference first
An arithmetic sequence adds the same number each time, while a geometric one multiplies by the same number. The quickest way to tell them apart is to try both on the first few terms: if the difference between neighbouring terms is constant, the sequence is arithmetic. If the ratio between neighbouring terms is constant, it is geometric. If neither is constant, it is something else — square numbers or Fibonacci, for instance.
The difference in growth is dramatic over time. An arithmetic sequence grows in a straight line and is easy to predict, while a geometric one grows exponentially and runs away from intuition fast. It is the same mechanism that underlies compound interest, and the reason a seemingly modest percentage growth per period ends in very large numbers after many periods.
Frequently asked questions
What's the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence, the same fixed number is added for each new term (for example +3 each time). In a geometric sequence, each term is multiplied by the same fixed number (for example ×3 each time), which gives much faster growth.
What happens to the sum of a geometric series if the ratio is between -1 and 1?
Then the terms get smaller and smaller each time, and the sum approaches a fixed limit even if you add up infinitely many terms. This calculator works out the sum of a finite number of terms n, not the infinite sum.
Where are number sequences used in practice?
Arithmetic sequences are used, among other things, to calculate fixed, even changes (like loan payments or saving a fixed amount), while geometric sequences are used for compound interest, population growth and other processes with a percentage change per period.