What is the nth term of a sequence?

The nth term, written a_n, is the value in position n of the sequence. A formula for it lets you jump straight to any term without listing all the earlier ones. For an arithmetic sequence a_n = a_1 + (n - 1)d, and for a geometric sequence a_n = a_1 * r^(n-1).

How do I find the sum of an arithmetic sequence?

Use S_n = n/2 * (a_1 + a_n), which is the number of terms times the average of the first and last term. If you do not know the last term a_n, use the equivalent form S_n = n/2 * [2a_1 + (n - 1)d]. For example, the first 10 terms of 3, 7, 11, 15, ... sum to 210.

How do I find the sum of a geometric sequence?

For a finite geometric series use S_n = a_1(1 - r^n) / (1 - r), valid for any common ratio r except r = 1. For example, the first 7 terms of 2, 6, 18, 54, ... sum to 2186. If r = 1, every term equals a_1 and the sum is simply n * a_1.

What is the difference between an arithmetic and a geometric sequence?

An arithmetic sequence adds a constant common difference d each step, so its terms plot as a straight line. A geometric sequence multiplies by a constant common ratio r each step, so its terms curve. Test it by checking whether consecutive terms have a constant difference (arithmetic) or a constant ratio (geometric).

Can this calculator find the common difference or common ratio from two terms?

This calculator works from the first term plus a known common difference d or common ratio r that you enter. To find d yourself, subtract consecutive terms (d = a_2 - a_1); to find r, divide consecutive terms (r = a_2 / a_1). Then enter that value to compute any term or sum.

Does the calculator handle infinite geometric series?

An infinite geometric series has a finite sum only when the absolute value of r is less than 1, given by S_infinity = a_1 / (1 - r). When the absolute value of r is at least 1 the series diverges and has no finite total. Use the formula directly for the infinite case, and the calculator for the nth term and any finite sum.

Sequence Calculator

Formula

Enter your values

Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

This sequence calculator finds the nth term and the sum of both arithmetic and geometric sequences, shows the working step by step, and plots the terms so you can see the pattern. Enter your values once and get the answer plus the formula behind it.

How to use this sequence calculator

The calculator handles both sequence types on this one page. Follow these steps:

Choose the sequence type — arithmetic (you add a fixed amount each step) or geometric (you multiply by a fixed amount each step).
Enter the first term, a₁ — the value the sequence starts from.
Enter the common difference d (arithmetic) or the common ratio r (geometric).
Enter the term position n — which term you want, e.g. n = 10 for the 10th term.
Press Calculate. The result panel shows the nth term and the sum of the first n terms.

Open the Steps tab to see the formula worked line by line, the Visual tab to plot the terms, and the Practice button to generate fresh problems to test yourself.

What is a sequence?

A sequence is an ordered list of numbers that follow a rule. Each number is called a term, and its place in the list is the term position, written n. The first term is a₁, the second is a₂, the third is a₃, and so on; the general term in position n is written aₙ. The nth term is just a formula that lets you jump straight to any position without listing everything before it.

This page covers the two most common types studied in algebra and pre-calculus. In an arithmetic sequence (also called an arithmetic progression) you add the same number each step — for example 3, 7, 11, 15, … adds 4 each time. In a geometric sequence (a geometric progression) you multiply by the same number each step — for example 2, 6, 18, 54, … multiplies by 3 each time.

Arithmetic sequence calculator

An arithmetic sequence grows by adding a constant value, called the common difference d, to each term. You find d by subtracting any term from the one after it: d = aₙ − aₙ₋₁. If that difference is the same everywhere, the sequence is arithmetic.

Arithmetic sequence formula

To find the term in position n directly, use the explicit nth-term formula:

aₙ = a₁ + (n − 1)d

Here a₁ is the first term, d is the common difference, and n is the position. You can also write the sequence recursively as aₙ = aₙ₋₁ + d with a₁ given — each term is the previous term plus d — but the explicit formula above is faster because it skips straight to term n.

Why n − 1: the first term needs zero steps of d, the second needs one step, the third needs two. So term n needs (n − 1) steps, not n.

Sum of an arithmetic sequence

The sum of the first n terms (called an arithmetic series and written Sₙ) has a clean closed form:

Sₙ = n⁄2 · (a₁ + aₙ)

This says the sum equals the number of terms times the average of the first and last term. If you do not yet know the last term aₙ, substitute the nth-term formula to get an equivalent version that uses only a₁, d, and n:

Sₙ = n⁄2 · [2a₁ + (n − 1)d]

Worked example — arithmetic (step by step)

Find the 10th term and the sum of the first 10 terms of 3, 7, 11, 15, …

Identify a₁ and d. The first term is a₁ = 3. The common difference is d = 7 − 3 = 4.
Apply the nth-term formula. a₁₀ = 3 + (10 − 1)(4) = 3 + 9·4 = 3 + 36 = 39.
Apply the sum formula. S₁₀ = 10⁄2 · (a₁ + a₁₀) = 5 · (3 + 39) = 5 · 42 = 210.
Check. The average term is (3 + 39)/2 = 21, and 10 terms × 21 = 210. The answers agree.

So the 10th term is 39 and the first ten terms add up to 210.

Geometric sequence calculator

A geometric sequence grows by multiplying each term by a constant value, called the common ratio r. You find r by dividing any term by the one before it: r = aₙ ⁄ aₙ₋₁. When that ratio is the same everywhere, the sequence is geometric. A ratio greater than 1 means growth; a ratio between −1 and 1 (but not 0) means the terms shrink toward zero.

Geometric sequence formula

The explicit nth-term formula for a geometric sequence is:

aₙ = a₁ · r⁽ⁿ⁻¹⁾

where a₁ is the first term, r is the common ratio, and n is the position. The exponent is again n − 1 for the same reason as before: the first term has been multiplied by r zero times. Working with these powers? Our exponent calculator and logarithm calculator help when you need to solve for n or r.

Sum of a geometric sequence

The sum of the first n terms of a geometric sequence (a finite geometric series) is:

Sₙ = a₁ · (1 − rⁿ) ⁄ (1 − r), r ≠ 1

This formula works for any common ratio except r = 1 (when every term equals a₁, so the sum is simply n·a₁). The expansion of these series connects directly to the binomial theorem calculator, where similar powered terms appear.

Infinite geometric series

If you keep adding terms forever, the result is an infinite geometric series. It only settles on a finite total — it converges — when the terms shrink, which happens precisely when |r| < 1. In that case rⁿ approaches 0 and the finite formula simplifies to:

S∞ = a₁ ⁄ (1 − r), |r| < 1

When |r| ≥ 1 the terms do not shrink, the partial sums grow without bound, and the series diverges — there is no finite sum.

Example: for 1 + ½ + ¼ + ⅛ + … we have a₁ = 1 and r = ½, so S∞ = 1 ⁄ (1 − ½) = 2.

Worked example — geometric (step by step)

Find the 7th term and the sum of the first 7 terms of 2, 6, 18, 54, …

Identify a₁ and r. The first term is a₁ = 2. The common ratio is r = 6 ÷ 2 = 3.
Apply the nth-term formula. a₇ = 2 · 3⁽⁷⁻¹⁾ = 2 · 3⁶ = 2 · 729 = 1458.
Apply the sum formula. S₇ = 2 · (1 − 3⁷) ⁄ (1 − 3) = 2 · (1 − 2187) ⁄ (−2) = 2 · (−2186) ⁄ (−2) = 2 · 1093 = 2186.
Check. Adding 2 + 6 + 18 + 54 + 162 + 486 + 1458 = 2186, which matches the formula.

So the 7th term is 1458 and the first seven terms add up to 2186.

Arithmetic vs geometric sequences

The two sequence types behave very differently. The table below puts them side by side so you can see exactly how each formula changes.

Feature	Arithmetic	Geometric
Pattern	Add the common difference d	Multiply by the common ratio r
nth-term formula	aₙ = a₁ + (n − 1)d	aₙ = a₁ · r⁽ⁿ⁻¹⁾
Sum of n terms	Sₙ = n⁄2 · (a₁ + aₙ)	Sₙ = a₁(1 − rⁿ) ⁄ (1 − r)
Example	3, 7, 11, 15, … (d = 4)	2, 6, 18, 54, … (r = 3)
Growth shape on the plot	Straight line (constant slope)	Curve (exponential growth or decay)

How to tell which one you have: subtract consecutive terms — if the difference is constant, it is arithmetic. Divide consecutive terms instead — if the ratio is constant, it is geometric.

Reading the plot of terms

The Visual tab plots each term against its position, and the shape tells you the type at a glance. An arithmetic sequence plots as points on a straight line, because adding the same d each step gives a constant slope. A geometric sequence curves: when |r| > 1 it bends sharply upward (exponential growth), and when |r| < 1 it flattens toward zero (exponential decay). If your plotted points fall on a line you have an arithmetic sequence; if they bend, it is geometric.

Common mistakes to avoid

Mistake 1 — Using n instead of n − 1Both nth-term formulas use the exponent or multiplier n − 1, not n, because the first term takes zero steps. For the 10th term, use (10 − 1) = 9, not 10.

Mistake 2 — Confusing d with rUse the common difference d (which you add) only for arithmetic sequences and the common ratio r (which you multiply) only for geometric ones. Mixing them gives a completely wrong sequence.

Mistake 3 — Forgetting r ≠ 1 in the finite sumThe formula Sₙ = a₁(1 − rⁿ)/(1 − r) divides by (1 − r), so it breaks when r = 1. If r = 1 every term equals a₁, and the sum is just n · a₁.

Mistake 4 — Trying to sum a divergent seriesThe infinite-sum formula S∞ = a₁/(1 − r) only applies when |r| < 1. If |r| ≥ 1 the series diverges and has no finite total, so the formula does not apply.

Mistake 5 — Reading d or r from a single pair carelesslyAlways confirm the pattern with more than one pair of terms. Check that the difference (or ratio) is the same across several consecutive terms before trusting d or r.

Studying for AP Precalculus? Arithmetic and geometric sequences open Unit 2 (Exponential and Logarithmic Functions) as the bridge into exponential growth. See the 2027 score calculator.

Frequently asked questions

What is the nth term of a sequence?

The nth term, written aₙ, is the value in position n of the sequence. A formula for it lets you jump straight to any term without listing all the earlier ones. For an arithmetic sequence aₙ = a₁ + (n − 1)d, and for a geometric sequence aₙ = a₁ · r⁽ⁿ⁻¹⁾.
How do I find the sum of an arithmetic sequence?

Use Sₙ = n⁄2 · (a₁ + aₙ), which is the number of terms times the average of the first and last term. If you do not know the last term aₙ, use the equivalent form Sₙ = n⁄2 · [2a₁ + (n − 1)d]. For example, the first 10 terms of 3, 7, 11, 15, … sum to 210.
How do I find the sum of a geometric sequence?

For a finite geometric series use Sₙ = a₁(1 − rⁿ) ⁄ (1 − r), valid for any common ratio r except r = 1. For example, the first 7 terms of 2, 6, 18, 54, … sum to 2186. If r = 1, every term equals a₁ and the sum is simply n · a₁.
What is the difference between an arithmetic and a geometric sequence?

An arithmetic sequence adds a constant common difference d each step, so its terms plot as a straight line. A geometric sequence multiplies by a constant common ratio r each step, so its terms curve. Test it by checking whether consecutive terms have a constant difference (arithmetic) or a constant ratio (geometric).
Can this calculator find the common difference or common ratio from two terms?

This calculator works from the first term plus a known common difference d or common ratio r that you enter. To find d yourself, subtract consecutive terms (d = a₂ − a₁); to find r, divide consecutive terms (r = a₂ ⁄ a₁). Then enter that value to compute any term or sum.
Does the calculator handle infinite geometric series?

An infinite geometric series has a finite sum only when |r| < 1, given by S∞ = a₁ ⁄ (1 − r). When |r| ≥ 1 the series diverges and has no finite total. Use the formula directly for the infinite case, and the calculator for the nth term and any finite sum.

Looking for related discrete-math tools? Try the combinations calculator, or browse all of our free math calculators.