Mathematics
Arithmetic and geometric sequences
Enter the first term, the common difference or ratio and how many terms: you get the nth term, the partial sum and the list of terms. Convergent geometric ones also show the infinite sum.
Adding and multiplying: two ways to advance
An arithmetic sequence advances by addition: each term is the previous one plus a constant, the common difference. A geometric sequence advances by multiplication: each term is the previous one times a constant. Everything else follows from that single difference, including the fact that the first grow linearly and the second exponentially — which is why a geometric sequence with even a modest ratio overtakes any arithmetic one.
The sum of the first n terms of an arithmetic sequence uses the trick attributed to Gauss: pair the first term with the last, the second with the second-to-last, and so on. Every pair gives the same sum and there are n/2 pairs, so the total is n times the average of the first and last terms. There is no need to add term by term even for a thousand of them.
The geometric sum has a different formula but an equally simple origin: multiply the sum by the ratio and subtract, and almost every term cancels, leaving two. When the ratio is smaller than one in absolute value the terms shrink steadily and the sum of infinitely many of them converges to a finite number — the result that makes perpetual annuities possible and resolves Zeno's paradoxes.
Common mistakes
- Using n instead of n−1 in the exponent or the multiple for the nth term: the first term corresponds to zero steps, not one. For the tenth term you add the difference nine times, not ten.
- Applying the geometric sum formula with a ratio of 1: the denominator vanishes, and the sum is simply n times the first term.
- Believing every infinite series converges: the sum to infinity exists only for geometric sequences with a ratio between −1 and 1. With a ratio of 2 the sum grows without bound.
Frequently asked questions
What is the formula for the nth term?
For an arithmetic sequence aₙ = a₁ + (n−1)·d. For a geometric one aₙ = a₁ · q^(n−1). In both the exponent or multiple is n−1, because there are n−1 steps from the first term to the nth.
How do you find the sum of the first n terms?
Arithmetic: Sₙ = n·(a₁ + aₙ)/2, the number of terms times the average of the first and last. Geometric: Sₙ = a₁·(1 − qⁿ)/(1 − q), valid for q different from 1.
When does a geometric series converge?
When the ratio is smaller than 1 in absolute value. The sum of infinitely many terms is then a₁/(1 − q). With a ratio of 1/2 and a first term of 1 the infinite sum is exactly 2.
What are geometric sequences used for in finance?
The future value of a capital at compound interest is a geometric sequence with ratio 1 + i, and the present value of an annuity is the sum of a geometric sequence. Perpetual annuities are precisely the convergent-series case.
How this calculation works
Arithmetic sequence with first term a₁ and common difference d: nth term aₙ = a₁ + (n−1)d; sum of the first n terms Sₙ = n(a₁ + aₙ)/2 = n[2a₁ + (n−1)d]/2. Geometric sequence with first term a₁ and ratio q: nth term aₙ = a₁·q^(n−1); sum of the first n terms Sₙ = a₁(1 − qⁿ)/(1 − q) for q ≠ 1, and Sₙ = n·a₁ for q = 1. Infinite geometric series: converges if |q| < 1 and equals S = a₁/(1 − q); diverges otherwise. Property: in an arithmetic sequence each term is the arithmetic mean of its neighbours; in a geometric one it is their geometric mean.