Arithmetic and Geometric Sequences: The Formulas Explained

What a numerical sequence is

Line up any set of numbers in order and you get a sequence. Among the number sets studied in school, a sequence becomes a progression when that order follows a fixed rule: not random, but generated by one operation repeated identically from term to term. There are two main families: the arithmetic sequence, where the operation linking one term to the next is addition, and the geometric sequence, where it’s multiplication.

The arithmetic sequence: the common difference that adds up

An arithmetic sequence keeps one quantity fixed: the gap between any two consecutive terms, called the common difference. In 5, 7, 9, 11, 13, 15 that gap is 2.

An nth term formula gets around having to write the sequence out one number at a time: a_n = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the position of the term being sought. Writing a rule with letters standing in for numbers is the same move studied in algebra: a formula that holds for any position, not just the one written out by hand.

Adding up an arithmetic sequence

Adding many arithmetic terms one by one isn’t necessary: S_n = n(a1+an)/2 does it in one move — the count of terms times the first plus the last, halved.

Worked example: in the sequence 2, 5, 8, 11, 14 (five terms, common difference 3), the sum works out like this: S_5 = 5 × (2+14) / 2 = 5 × 16 / 2 = 40. There’s no need to add the five numbers one by one: the first term, the last term, and the count are enough.

The geometric sequence: the common ratio that multiplies

In a geometric sequence the numbers are never zero, and moving from one term to the next means multiplying by the same fixed number every time: the common ratio. In the sequence 2, 6, 18, 54 the common ratio is 3, because each term is the previous one multiplied by 3.

The nth term of a geometric sequence is a_n = a1 · q^(n-1), with a1 the first term and q the common ratio. Depending on the value of q, the sequence represents exponential growth, when q is above 1, or exponential decay, when q sits between 0 and 1: the same behavior found when studying exponentials and logarithms.

Adding up a geometric sequence, finite and infinite

With a common ratio different from 1, the total of the first n terms of a geometric sequence is given by S_n = a1 · (1-q^n) / (1-q).

But keep q between -1 and 1, and something happens that can never happen in an arithmetic sequence, no matter how small its common difference gets: even adding infinitely many terms, one after another forever, without ever stopping, the result stays a finite number rather than growing without bound. In that case the sum of the whole sequence equals a1 / (1-q), because each new term is smaller than the one before it, and the running total keeps approaching a ceiling it never quite reaches. That’s the calculation behind, for example, the present value of a perpetuity, a payment that repeats every year forever: an infinite stream of future money that still adds up, today, to a single finite figure.

When the ratio is above 1: the wheat and chessboard problem

The opposite case, a ratio above 1, produces growth that quickly outruns everyday intuition. The classic problem shows it well: place 1 grain of wheat on the first square of a chessboard and double it on every square after that, up to the sixty-fourth. The earliest known written version of this story dates to 1256, attributed to Ibn Khallikan.

The total across the 64 squares is 2^64 - 1, that is 18,446,744,073,709,551,615: a number that a formula writes out in seconds, but that no one could ever count by hand. It’s a special case of the same geometric sum seen above, with first term 1 and ratio 2, and it’s still a direct way to show just how fast geometric sequences grow once the ratio climbs above 1.

The same kind of growth, though much slower, governs compound interest: interest isn’t calculated only on the initial capital but also on whatever has already accrued in earlier periods, and year after year the added interest starts earning interest of its own. The same mechanism, applied to the cost of a loan such as a mortgage, is what determines its effective annual rate.

Arithmetic versus geometric: same structure, a different operation

The two families of sequences share a scaffolding — a first term, a common difference or ratio, an nth term formula, a sum — but the operation linking one term to the next changes everything about how the sequence behaves.

Aspect	Arithmetic sequence	Geometric sequence
Operation between a term and the next	add the common difference	multiply by the common ratio
Example	5, 7, 9, 11, 13, 15 (common difference 2)	2, 6, 18, 54 (common ratio 3)
nth term	a_n = a1 + (n-1)d	a_n = a1 · q^(n-1)
Sum of the first n terms	S_n = n(a1+an)/2	S_n = a1(1-q^n)/(1-q), with q ≠ 1
Sum of infinitely many terms	no finite value exists	finite if the absolute value of q is below 1

Anyone who has already met another recursive rule, the Fibonacci sequence, finds a third case useful for comparison: there, each term depends not on a single fixed common difference or ratio, but on the sum of the two previous terms — neither a constant addition nor a constant multiplication.