Number Sequence Calculator

How to use the number sequence calculator

1. Choose the type. An arithmetic sequence adds the same number each time (5, 8, 11, 14, …). A geometric sequence multiplies by the same number (2, 10, 50, 250, …).
2. Enter the first term, the step and n. The step is the common difference d for arithmetic sequences or the common ratio r for geometric ones. n is the term you want, counting the first term as n = 1.
3. Or paste some terms. Type the first three or more terms into “Identify from terms”. If they follow either pattern, a button fills in a₁ and d or r for you.

The calculator opens on 5, 8, 11, 14, … from OpenStax’s Precalculus 2e, section 11.4. Its 10th term is 32, and the first 10 terms add up to 185.

Arithmetic sequences

In an arithmetic sequence, the difference between consecutive terms is constant. Subtract any term from the next one to find it. In section 11.2, Example 1, the sequence −3, 1, 5, 9, 13 has a common difference of 4, while 1, 2, 4, 8, 16 is not arithmetic, since its differences are 1, 2, 4 and 8.

The explicit formula gives any term directly:

For 2, 12, 22, 32, 42, … (Example 5), a₁ = 2 and d = 10, so aₙ = 2 + 10(n − 1) = 10n − 8. The 100th term is 992.

The recursive formula describes each term from the one before: aₙ = aₙ₋₁ + d, starting from a₁. It is how you would list terms by hand, and it appears on the “Recursive rule” card.

The formula can also be solved for n. In Example 6, the finite sequence 8, 1, −6, …, −41 has d = −7, and −41 = 8 + (n − 1)(−7) gives n = 8 terms.

Geometric sequences

In a geometric sequence, the ratio of consecutive terms is constant. Divide any term by the one before it. Section 11.3, Example 1, shows that 1, 2, 4, 8, 16 has a common ratio of 2, while 48, 12, 4, 2 is not geometric, since its ratios are 1/4, 1/3 and 1/2.

For 2, 10, 50, 250, … (Example 5), aₙ = 2 × 5ⁿ⁻¹. With a negative ratio the signs alternate: a₁ = 5 and r = −2 give 5, −10, 20, −40 (Example 2). A ratio between −1 and 1 makes the terms shrink toward 0, as in 48, 24, 12, 6, … with r = 1/2.

Geometric sequences describe repeated percentage change. A population that grows 4% a year is multiplied by r = 1.04 each year, so after n years it is the starting value times 1.04ⁿ, the setup of the textbook’s Example 6.

Sums of the first n terms

Adding the terms of a sequence gives a series. Section 11.4 gives a formula for each type.

Arithmetic: Sₙ = n(a₁ + aₙ) ÷ 2, the number of terms times the average of the first and last. For 5 + 8 + … + 32, S₁₀ = 10 × (5 + 32) ÷ 2 = 185. For 20 + 15 + 10 + … + (−50), there are 15 terms and the sum is 15 × (20 − 50) ÷ 2 = −225.

In the textbook’s Example 3, someone walks half a mile on the first Sunday and a quarter mile more each week. Over 8 weeks that is an arithmetic series with a₁ = 0.5 and d = 0.25, totaling 11 miles.

Geometric: Sₙ = a₁(1 − rⁿ) ÷ (1 − r), for r ≠ 1. For 8 − 4 + 2 − …, r = −1/2 and S₁₁ ≈ 5.336. For Σ 3 × 2ᵏ from k = 1 to 6, the first term is 6, r = 2, and the sum is 378. When r = 1, every term equals a₁, so the sum is n × a₁.

Infinite geometric series

An arithmetic series with d ≠ 0 grows without limit, so it has no infinite sum. A geometric series does have one when the ratio is between −1 and 1, because the terms shrink fast enough:

In Example 7, 248.6 + 99.44 + 39.776 + … has r = 0.4, so its sum is 248.6 ÷ 0.6 ≈ 414.33. The series 4,374 × (−1/3)ᵏ⁻¹ sums to 4,374 ÷ (4/3) = 3,280.5. The calculator shows “None” when |r| is 1 or more.

The same idea turns a repeating decimal into a fraction. Example 8 writes 0.333… as 3/10 + 3/100 + 3/1,000 + …, a geometric series with a₁ = 0.3 and r = 0.1, whose sum is 0.3 ÷ 0.9 = 1/3.

Identifying a sequence

Paste three or more terms into “Identify from terms”. The calculator checks whether all the differences match (arithmetic) or all the ratios match (geometric) and, if so, reports a₁ and the step. Sequences that are neither, such as the squares 1, 4, 9, 16 or the Fibonacci numbers 1, 1, 2, 3, 5, are reported as such; their next terms follow other rules.

A few terms can fit more than one rule, so a pattern found from three terms is a good guess, not proof. Where a problem states the type or gives the formula, go by that.

Large and small results

Geometric terms grow quickly: 2 × 5⁹⁹ has 70 digits. Results that large switch to scientific notation, as do very small ones, and are accurate to about 15 significant digits. For exact work with whole numbers, the exponent calculator gives every digit of a single power.

Sequences in everyday problems

Arithmetic sequences show up wherever something grows by a fixed amount. A theater with 20 seats in the first row and 2 more in each row behind it has 20 + 2 × 14 = 48 seats in row 15, and 15 × (20 + 48) ÷ 2 = 510 seats in the first 15 rows. A stack of cans with 1 can on top and 1 more in each layer below holds 1 + 2 + … + n cans in n layers.

Geometric sequences describe repeated multiplication: a quantity that doubles each step, a sheet of paper folded in half again and again, or a value that loses the same share each period. Folding paper doubles the layers each time, so after 7 folds there are 2⁷ = 128 layers, which is why folding a sheet many times gets hard so quickly.

The quick test: subtract neighboring terms. If the differences match, the sequence is arithmetic. If they don’t, divide neighboring terms; matching ratios mean it is geometric. If neither works, the sequence follows some other rule, and the “Identify from terms” box says so.

Sources & assumptions

How this calculator gets its numbers. Full method: our methodology.

• Arithmetic: aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ) ÷ 2.
• Geometric: aₙ = a₁rⁿ⁻¹ and Sₙ = a₁(1 − rⁿ) ÷ (1 − r), or n × a₁ when r = 1; the infinite sum a₁ ÷ (1 − r) exists only when −1 < r < 1.
• Terms are numbered from n = 1.

Sources

1. OpenStax (Rice University), Precalculus 2e, 11.2 Arithmetic Sequences. Common difference, aₙ = a₁ + (n − 1)d and recursive formulas (Examples 1–6).
2. OpenStax (Rice University), Precalculus 2e, 11.3 Geometric Sequences. Common ratio and aₙ = a₁rⁿ⁻¹ (Examples 1–5).
3. OpenStax (Rice University), Precalculus 2e, 11.4 Series and Their Notations. Arithmetic and geometric partial sums and the infinite geometric sum a₁ ÷ (1 − r) for |r| < 1 (Examples 2–7).

Last reviewed November 2, 2026. For a single power such as rⁿ, use the exponent calculator.