Geometric Sequence Calculator
Calculate the nth term, common ratio, explicit formula, finite series sum, and infinite-series result for geometric sequences, including growth, decay, and alternating-sign patterns.
What makes a sequence geometric?
A geometric sequence changes by multiplication. Instead of adding the same difference every time, each term is multiplied by the same common ratio. A sequence like 3, 6, 12, 24 is geometric because each term is multiplied by 2. A sequence like 80, 40, 20, 10 is also geometric because each term is multiplied by 0.5.
OpenStax presents the nth-term rule as an explicit formula in its College Algebra section on geometric sequences. This calculator uses that same idea, then adds the finite series sum and infinite-series check because students often need both on the same assignment.
Geometric sequence and series formulas
The infinite sum formula only works when the terms shrink toward zero. OpenStax explains the finite geometric sum in its geometric sequences and series lesson, including the idea that the sum is built from repeated powers of the common ratio.
How to choose between arithmetic and geometric
If the difference is constant, the sequence is arithmetic. If the ratio is constant, the sequence is geometric. The sequence 5, 10, 15, 20 has equal differences, so it is arithmetic. The sequence 5, 10, 20, 40 has equal ratios, so it is geometric. Students often confuse these when both sequences are increasing, but the test is simple: subtraction checks arithmetic; division checks geometric.
| Pattern | Test | Example | Formula type |
|---|---|---|---|
| Arithmetic | Subtract terms | 4, 7, 10, 13 | aₙ = a₁ + (n - 1)d |
| Geometric | Divide terms | 4, 8, 16, 32 | aₙ = a₁rⁿ⁻¹ |
Worked example: growth pattern
The term grows much faster than an arithmetic sequence because each step multiplies the previous value. That is why geometric sequences appear in compound growth, repeated doubling, population models, depreciation, and half-life style problems.
Important edge cases
If r = 1, every term is the same, so the finite sum is simply n times the first term. If r = 0, every term after the first becomes zero. If r is negative, the sequence alternates signs: positive, negative, positive, negative. For infinite sums, the common ratio must be between -1 and 1. Otherwise, the terms do not settle down toward zero, and the infinite series does not have a finite total.
Common questions
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Use a_n = a_1r^(n - 1). The first term is a_1, the common ratio is r, and n is the term position you want. The exponent is n - 1 because the first term already exists before any multiplication happens. For example, in 3, 6, 12, 24, the first term is 3 and the ratio is 2, so the 7th term is 3 × 2^6 = 192.
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Divide a term by the term before it. For example, in 5, 15, 45, 135, the common ratio is 15 ÷ 5 = 3, and the same result appears when you check 45 ÷ 15 or 135 ÷ 45. If the ratios are not all the same, the sequence is not geometric. Always check more than one pair if the problem gives several terms.
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A geometric sequence is the list of terms. A geometric series is the sum of those terms. For example, 2, 4, 8, 16 is a geometric sequence, while 2 + 4 + 8 + 16 is a geometric series. A question asking for 'the 10th term' needs the sequence formula. A question asking for 'the sum of the first 10 terms' needs the series formula.
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An infinite geometric series has a finite sum only when the common ratio has an absolute value less than 1. That means -1 < r < 1. In that case, the terms shrink toward zero, so the total approaches a fixed number. If r is 2, the terms grow. If r is -2, the terms grow while alternating signs. In both cases, there is no finite infinite sum.
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Yes. A negative common ratio makes the signs alternate. For example, 3, -6, 12, -24 is geometric with r = -2. The nth-term formula still works, but you must keep the negative sign inside the power. The sign of the answer depends on whether n - 1 is even or odd.
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If the common ratio is a fraction between 0 and 1, the sequence decreases toward zero. For example, 80, 40, 20, 10 has r = 1/2. This is common in half-life, repeated discount, and depreciation problems. The terms get smaller, but they do not become exactly zero unless the ratio itself is zero.
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The first term is already the starting value, so no multiplication is needed to produce a_1. To reach a_2, you multiply once. To reach a_3, you multiply twice. Therefore, to reach a_n, you multiply n - 1 times. Using n instead of n - 1 gives the next term after the one you were trying to find.
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Zero can appear in a geometric sequence, but it creates special cases. If the first term is nonzero and the common ratio is zero, all terms after the first are zero. If a sequence begins with zero, the ratio may be undefined because division by zero is not allowed. For most school-level geometric sequence problems, the given terms are nonzero so the common ratio can be found by division.