A geometric sequence changes by a constant multiplicative factor rather than a constant additive difference. Once the first term and common ratio are known, every later term follows from repeated multiplication. This calculator also derives the sequence from two consecutive indexed terms, where the ratio is uniquely determined whenever the data are valid.
How to use the Geometric Sequence Calculator
If you know the first term and common ratio, choose First term + common ratio. Enter a₁, r, and the requested index n. The calculator returns the nth term, explicit rule, sum of the first n terms, a sequence preview, and whether the corresponding infinite series converges.
If your problem gives two consecutive terms such as a₄ = 54 and a₅ = 162, choose Two consecutive known terms. The calculator divides the later term by the earlier term to obtain r, then works backward to recover a₁.
Values may be integers, terminating decimals, or fractions such as 5/2 or −1/3. Indices are positive whole numbers because this page uses the standard a₁-first convention.
What is a geometric sequence?
A geometric sequence is one in which each term after the first is obtained by multiplying the preceding term by the same constant ratio r.
r = a₂/a₁ = a₃/a₂ = a₄/a₃ = ...For 3, 6, 12, 24, ... the common ratio is 2. For 81, 27, 9, 3, ... it is 1/3. For 5, −10, 20, −40, ... it is −2.
The defining pattern is multiplication. That separates geometric sequences from arithmetic sequences, which use a constant difference instead.
The nth-term formula
To move from a₁ to a₂, multiply by r once. To reach a₃, multiply by r twice. In general, reaching the nth term requires n−1 multiplications:
aₙ = a₁rⁿ⁻¹For a₁ = 4 and r = 3, the sixth term is a₆ = 4(3⁵) = 972.
The exponent n−1 is essential. Using rⁿ instead would shift the entire sequence by one term.
Finding the ratio from two consecutive known terms
When two consecutive nonzero terms are known, the ratio is simply:
r = aₖ₊₁ / aₖOnce r is known, use aₖ = a₁rᵏ⁻¹ to solve backward for a₁:
a₁ = aₖ / rᵏ⁻¹Using consecutive terms has an important advantage over using two arbitrary separated terms. If only a₂ and a₄ are known, solving r² = a₄/a₂ may produce two possible real ratios when the quotient is positive. Consecutive terms avoid that root ambiguity.
If a known term is zero, special care is required. In a standard geometric sequence, once a nonzero sequence reaches zero through r = 0, every later term remains zero. A pair 0 followed by a nonzero value cannot belong to one geometric sequence.
Sum of the first n terms
The terms of a finite geometric series are:
Sₙ = a₁ + a₁r + a₁r² + ... + a₁rⁿ⁻¹For r ≠ 1, multiplying the sum by r and subtracting causes the interior terms to cancel, giving:
Sₙ = a₁(1 − rⁿ)/(1 − r)An equivalent version is a₁(rⁿ−1)/(r−1). Both formulas produce the same result.
When r = 1, every term equals a₁, so division by 1−r would be invalid. In that special case, Sₙ = na₁.
When does an infinite geometric series converge?
A geometric sequence and an infinite geometric series are related but not identical: the sequence lists terms, while the series adds them. An infinite geometric series has a finite sum only when the terms shrink rapidly enough toward zero.
If |r| < 1, then S∞ = a₁/(1 − r)If |r| ≥ 1, the standard infinite series does not converge to a finite value. With r = 1 the terms stay constant; with r = −1 they alternate without settling; and with |r| > 1 their magnitudes grow.
A negative ratio can still converge. For example, r = −1/2 has |r| = 1/2, so the alternating terms shrink and the infinite series converges.
Understanding different common ratios
r > 1: magnitudes grow when a₁ is nonzero and positive r preserves sign.
0 < r < 1: terms keep the same sign and shrink toward zero.
r < 0: signs alternate from term to term. If |r| > 1, magnitudes grow; if |r| < 1, magnitudes shrink.
r = 1: the sequence is constant.
r = 0: the first term is followed by zeros: a₁, 0, 0, 0, ... .
The common ratio itself does not have to be a whole number. Fractions, negative values, and decimals are all valid.
Fractions, decimals, and exact geometric values
Fractional ratios are extremely common. The sequence 64, 32, 16, 8, ... has r = 1/2. Exact fractions make the structure easier to see than rounded decimals.
SonoCalculator preserves rational values exactly for integer, terminating-decimal, and fraction inputs when the resulting powers remain practical for browser calculation. A ratio of 1/3 stays exactly 1/3 rather than being replaced by 0.3333 before exponentiation.
Because powers can grow extremely quickly, the calculator limits requested indices to a practical range. That is a browser safeguard, not a mathematical limit on geometric sequences.
Connection to exponential functions
The explicit rule aₙ = a₁rⁿ⁻¹ has an exponential form because the index appears in the exponent. A geometric sequence can therefore be viewed as discrete samples from a related exponential function.
When r > 1, this resembles exponential growth. When 0 < r < 1, it resembles exponential decay. A negative ratio produces alternating signs, which is not represented by an ordinary positive-base real exponential curve in the same simple way.
This differs from an arithmetic sequence, whose nth-term rule is linear in n. Arithmetic patterns add a constant amount; geometric patterns multiply by a constant factor.
Worked geometric-sequence examples
Example 1: growing sequence
Let a₁ = 3, r = 2, and n = 8:
a₈ = 3·2⁷ = 384S₈ = 3(1−2⁸)/(1−2) = 765.
Example 2: decay ratio
For a₁ = 64 and r = 1/2:
a₈ = 64(1/2)⁷ = 1/2Because |r| < 1, the infinite series converges to 64/(1−1/2) = 128.
Example 3: negative ratio
Let a₁ = 5 and r = −2:
5, −10, 20, −40, 80, ...The signs alternate and the magnitudes double. The infinite series does not converge because |r| = 2.
Example 4: solve from consecutive terms
If a₄ = 54 and a₅ = 162, then r = 162/54 = 3. Since a₄ = a₁·3³:
a₁ = 54/27 = 2The sequence begins 2, 6, 18, 54, 162, ... .
Example 5: r = 1
If a₁ = 7 and r = 1, every term is 7. Therefore a₂₀ = 7 and S₂₀ = 140.
Common geometric-sequence mistakes
Using a constant difference instead of a ratio. Equal differences describe arithmetic sequences, not geometric ones.
Using exponent n instead of n−1. The first term contains zero factors of r beyond a₁.
Applying the finite-sum formula with r = 1. Use Sₙ = na₁ instead.
Assuming every infinite geometric series has a finite sum. Convergence requires |r| < 1.
Ignoring negative ratios. A negative r is valid and produces alternating signs.
Rounding a fractional ratio too early. Powers amplify rounding errors, so exact fractions are preferable when possible.
Trying to infer a unique ratio from widely separated terms without checking roots. Nonconsecutive terms can create multiple-ratio possibilities. Consecutive terms avoid this ambiguity.
Frequently asked questions
How do I find the nth term of a geometric sequence?
Use aₙ = a₁rⁿ⁻¹, where a₁ is the first term and r is the common ratio.
How do I find the common ratio?
Divide any nonzero term by the preceding term. For consecutive terms, r = aₖ₊₁/aₖ.
How do I find the sum of the first n terms?
For r ≠ 1, use Sₙ = a₁(1−rⁿ)/(1−r). For r = 1, use Sₙ = na₁.
When does an infinite geometric series converge?
It converges to a finite sum when |r| < 1. The sum is a₁/(1−r).
Can the common ratio be negative?
Yes. A negative common ratio causes the terms to alternate signs.
Can the common ratio be zero?
Yes. Starting from a₁, every later term becomes zero.
Can a geometric sequence contain fractions?
Yes. Both a₁ and r can be fractional, and fractional ratios are common in decay sequences.
Why does this calculator use consecutive terms instead of any two arbitrary terms?
Two separated terms can require solving a power equation for r, which may have multiple real solutions or no real solution. Consecutive terms provide a direct ratio and avoid that ambiguity.
Important note
This calculator assumes one geometric sequence indexed from a₁ with a single constant common ratio. Infinite-series output is shown as a finite sum only when |r| < 1. If observed ratios vary, or if separated known terms allow multiple possible ratios, additional information is needed before a unique geometric sequence can be identified.