🔢 Geometric Sequence Calculator: nth Term, Sum and Common Ratio

What is the Geometric Sequence Calculator?

A geometric sequence, also called a geometric progression, starts with a first term a and builds each later term by multiplying by a constant ratio r. The terms are a, a x r, a x r^2, a x r^3, and so on. The single property that makes a sequence geometric is that the ratio between any two consecutive terms is always the same: divide any term by the one immediately before it and you always get r. That is the test for whether a list of numbers is geometric at all. For 5, 15, 45, 135 you get 15/5 = 3, 45/15 = 3 and 135/45 = 3, so the sequence is geometric with r = 3. For 5, 15, 40 you get 3 and then about 2.67, so it is not geometric.

Two formulas do most of the work, and textbooks call them the explicit and the recursive forms. The explicit formula for the nth term is a_n = a x r^(n-1), because to reach the nth term you multiply the first term by r a total of (n-1) times. It jumps straight to any position: the 20th term needs no knowledge of the 19th. The recursive formula is a_n = r x a_(n-1) with a_1 = a, which describes the same sequence one step at a time and is how spreadsheets and computer code usually build it. Use the explicit form when you want a distant term fast, and the recursive form when you are generating the whole list or writing the rule from a pattern you can see.

The sum of the first n terms, called a finite geometric series, is S_n = a x (1 - r^n) / (1 - r) when r is not 1. An equivalent version, a x (r^n - 1) / (r - 1), gives exactly the same answer and is tidier when r is bigger than 1 because it avoids negative numbers in both the top and the bottom. When r equals 1 every term is just a, so the sum is simply S_n = a x n and the general formula must not be used, since 1 - r would be zero. This calculator switches to a x n automatically at r = 1, which is a case many quick online calculators get wrong or refuse outright.

There is also an infinite sum. If the common ratio sits strictly between -1 and 1, so the size of r is less than 1, the terms shrink toward zero fast enough that adding infinitely many of them still gives a finite answer: S = a / (1 - r). This is the standard convergence condition stated in OpenStax Precalculus and every algebra course, and it comes straight from the finite formula: as n grows, r^n heads to zero, so a x (1 - r^n) / (1 - r) settles on a / (1 - r). If the size of r is 1 or larger, the terms do not shrink, the partial sums keep growing or keep flipping between two values, and the series diverges, so no finite infinite sum exists. This calculator reports the infinite sum only when the series genuinely converges, and says so plainly when it does not.

Two more jobs come up constantly and are worth knowing because the calculator handles them once you rearrange by hand. First, finding the common ratio from two non-adjacent terms: if you know a_1 = 3 and a_4 = 24, then a_4 = a_1 x r^3, so 24 = 3 x r^3, r^3 = 8 and r = 2. In general, if you know two terms a_p and a_q, then r = (a_q / a_p)^(1 / (q - p)). Second, the geometric mean: the number sitting between two terms of a geometric sequence is the square root of their product, so the term between 4 and 25 is the square root of 100, which is 10. That is the geometric mean, and it is why growth rates are averaged geometrically rather than arithmetically.

Geometric sequences are not just exam material. Compound interest is a geometric sequence with r = 1 + the interest rate per period, so a balance of 1000 at 5 percent per year runs 1000, 1050, 1102.50 and the nth year balance is 1000 x 1.05^(n-1). Radioactive decay and drug half-lives are geometric with r = 0.5. Depreciation on a reducing balance uses r = 1 minus the depreciation rate. Even the classic paradox of a ball bouncing to 60 percent of its previous height forever, yet travelling a finite total distance, is nothing more than a convergent geometric series with r = 0.6. Recognising the pattern is usually the hard part; once you have a and r, the formulas above finish the job in one line.

When to use it

• Checking homework on geometric progressions: finding a missing nth term, the common ratio, or a finite series sum.
• Modelling compound growth or decay, where a quantity is multiplied by the same factor each period, such as interest, inflation, or reducing balance depreciation.
• Working out the total of a repeating halving or doubling pattern, such as a bouncing ball, a paper folding problem, or a half-life chain.
• Confirming whether an infinite geometric series converges and, if so, what value it adds up to.
• Converting between the explicit formula a_n = a x r^(n-1) and the recursive formula a_n = r x a_(n-1) for a sequence you can see.
• Generating the first n terms quickly to spot a pattern, fill a table, or sanity check a spreadsheet model.

How to use the Geometric Sequence Calculator

1. Enter the first term (a), the starting value of the sequence. It can be negative or a decimal.
2. Enter the common ratio (r), the fixed multiplier between consecutive terms. If you do not know r, divide any term by the term before it.
3. Enter the number of terms (n) you want to include. This is also the position used for the nth term.
4. Read off the nth term, the sum of the first n terms, and the full term list.
5. Check the infinite sum panel. It shows a value only when the size of r is less than 1; otherwise the series diverges and no finite total exists.

Formula & method

Worked examples

Behaviour of a geometric sequence by common ratio r

Example sequence with a = 3, r = 2

Geometric versus arithmetic sequences at a glance

Common mistakes to avoid

• Using n instead of n-1 in the nth term formula. The exponent is n-1, not n, because the first term has been multiplied by r zero times. With a = 2 and r = 3, the 5th term is 2 x 3^4 = 162, not 2 x 3^5 = 486. This single off by one error is the most common wrong answer on geometric sequence questions.
• Confusing a geometric sequence with an arithmetic one. In a geometric sequence you multiply by a fixed ratio; in an arithmetic sequence you add a fixed difference. The pattern 2, 6, 18, 54 is geometric (times 3), while 2, 6, 10, 14 is arithmetic (plus 4). Test by dividing consecutive terms: if the quotient is constant it is geometric, if the difference is constant it is arithmetic.
• Expecting an infinite sum when r is too large. The infinite sum a / (1 - r) only exists when the size of r is strictly less than 1. If r is 1, -1, or bigger in size, the series diverges and has no finite total. Plugging r = 2 into a / (1 - r) produces a negative number that looks like an answer but is meaningless.
• Dividing by zero when r = 1. The sum formula a x (1 - r^n) / (1 - r) breaks when r = 1 because 1 - r = 0. When every term equals a, just use S_n = a x n instead. A sequence like 7, 7, 7, 7 is technically geometric with r = 1, and the sum of 4 terms is simply 28.
• Losing the negative sign when r is negative. With a negative ratio the terms alternate in sign, and r^(n-1) is positive for odd n and negative for even n. For a = 5 and r = -2 the terms are 5, -10, 20, -40. Bracket the ratio when you compute powers: (-2)^3 = -8, but -2^3 read as minus 2 cubed is also -8 while (-2)^2 = 4 and -2^2 = -4, which is where sign errors creep in.
• Taking only the positive root when solving for r. If a_1 = 2 and a_3 = 18, then r^2 = 9 gives r = 3 or r = -3, and both produce valid geometric sequences (2, 6, 18 and 2, -6, 18). Even powers leave two answers. Only an extra piece of information, such as a stated sign or another term, picks one.

Glossary

Geometric sequence

A list of numbers where each term is the previous term multiplied by a fixed common ratio. Also called a geometric progression.

First term (a)

The starting value of the sequence, the term you multiply by the ratio to build the rest. Often written a_1.

Common ratio (r)

The fixed number that each term is multiplied by to get the next term. Found by dividing any term by the one before it.

nth term

The value at position n in the sequence, found with the explicit formula a_n = a x r^(n-1).

Explicit formula

A rule that gives any term directly from its position n, without needing the previous term: a_n = a x r^(n-1).

Recursive formula

A rule that defines each term from the one before it: a_n = r x a_(n-1), with a_1 = a as the starting point.

Geometric series

The sum of the terms of a geometric sequence, either a finite number of them (S_n) or infinitely many (S).

Convergence

When an infinite series adds up to a finite value, which happens for a geometric series only if the size of r is less than 1.

Geometric mean

The term that sits between two numbers in a geometric sequence, equal to the square root of their product. Between 4 and 25 it is 10.

Frequently asked questions

Sources

• Geometric Sequences, OpenStax Precalculus 1e, Section 11.3 , LibreTexts Mathematics
• Explicit Formulas for Geometric Sequences, College Algebra , Lumen Learning