Arithmetic and Geometric Sequence Calculator

What Is a Sequence?

A sequence is simply an ordered list of numbers, called terms, that follow some consistent rule from one term to the next. The position of a term in the list is usually written as a subscript — a₁ is the first term, a₂ is the second term, and a_n is the "nth" term, meaning whichever position you're interested in. Sequences can be finite (a fixed number of terms) or infinite (continuing forever), and the rule connecting consecutive terms is what determines the type of sequence. The two most common types — and the two this calculator handles — are arithmetic sequences, which add a constant amount each step, and geometric sequences, which multiply by a constant amount each step.

Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always the same. That constant amount is called the common difference, usually written as d. To get the next term, you simply add d to the current term.

Consider the sequence 3, 8, 13, 18… Each term is 5 more than the one before it, so the common difference is d = 5, and the first term is a₁ = 3. The formula for any term in an arithmetic sequence is:

a_n = a₁ + (n − 1)d

To find the 10th term of 3, 8, 13, 18…, substitute a₁ = 3, d = 5, and n = 10: a₁₀ = 3 + (10 − 1) × 5 = 3 + 45 = 48. Notice that d can also be negative, which produces a sequence that counts down instead of up (like 20, 15, 10, 5, 0, −5…), and d can be zero, which produces a constant sequence where every term is identical.

Geometric Sequences

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same fixed number, called the common ratio, usually written as r. Instead of adding a constant amount, geometric sequences grow (or shrink) by a constant factor.

Consider the sequence 2, 6, 18, 54… Each term is 3 times the one before it, so the common ratio is r = 3 and the first term is a₁ = 2. The formula for any term in a geometric sequence is:

a_n = a₁ × r^{(n − 1)}

To find the 6th term of 2, 6, 18, 54…, substitute a₁ = 2, r = 3, and n = 6: a₆ = 2 × 3⁵ = 2 × 243 = 486. A negative common ratio produces an alternating sequence that flips sign every term (like 4, −8, 16, −32…), and a ratio between −1 and 1 (but not zero) produces a sequence that shrinks toward zero rather than growing.

Sum of a Sequence (Series)

When you add up the terms of a sequence, the result is called a series, and the sum of the first n terms is written S_n. Both sequence types have a shortcut formula that avoids adding every term one at a time.

For an arithmetic sequence, the sum of the first n terms is:

S_n = n/2 × (2a₁ + (n − 1)d)

This works because pairing the first term with the last, the second term with the second-to-last, and so on always produces the same pair-sum, and there are n/2 such pairs.

For a geometric sequence (with r not equal to 1), the sum of the first n terms is:

S_n = a₁(1 − rⁿ) / (1 − r)

If r = 1 every term is identical to a₁, so the formula above would divide by zero; in that special case the sum is simply S_n = n × a₁.

Geometric series have one more trick that arithmetic series don't: when the common ratio's absolute value is less than 1 (in other words, −1 < r < 1), each new term gets smaller and smaller, and the sum of infinitely many terms converges to a finite number rather than growing without bound. That sum-to-infinity is:

S_∞ = a₁ / (1 − r), valid only when |r| < 1

If |r| ≥ 1, the terms never shrink, so adding infinitely many of them never settles on a finite total — the sum-to-infinity formula simply doesn't apply in that case.

Frequently Asked Questions

What's the difference between a sequence and a series?

A sequence is the ordered list of individual terms itself — for example, 3, 8, 13, 18. A series is what you get when you add those terms together — 3 + 8 + 13 + 18. In short, a sequence is a list, and a series is a sum.

Can the common ratio be negative?

Yes. A negative common ratio is perfectly valid and produces an alternating sequence — the sign flips back and forth every term because multiplying by a negative number reverses sign each time. For example, with a₁ = 4 and r = −2, the sequence runs 4, −8, 16, −32, 64…, growing in magnitude but alternating between positive and negative.

What happens to the sum if r = 1?

The standard geometric sum formula, S_n = a₁(1 − rⁿ)/(1 − r), has (1 − r) in the denominator, so when r = 1 that denominator becomes zero and the formula breaks down. But r = 1 just means every term equals a₁, so the sum of n identical terms is simply n × a₁ — no special formula needed, just repeated addition.

How do I tell if a sequence is arithmetic or geometric just by looking at it?

Check consecutive terms with subtraction and with division. If subtracting each term from the next always gives the same number, it's arithmetic (that number is d). If dividing each term by the previous one always gives the same number, it's geometric (that number is r). Some sequences are neither — for example, 1, 4, 9, 16 (perfect squares) has neither a constant difference nor a constant ratio.

Why does the geometric series sum-to-infinity formula only work when |r| < 1?

The formula S_∞ = a₁/(1 − r) comes from taking the limit of S_n = a₁(1 − rⁿ)/(1 − r) as n grows infinitely large. That limit only settles on a finite value if rⁿ shrinks toward zero as n increases, which only happens when |r| < 1. If |r| ≥ 1, rⁿ either stays at a fixed size (r = 1 or r = −1) or grows without bound (|r| > 1), so the running total never approaches a fixed number — it either grows forever or oscillates without settling down.

Can n be a decimal or a fraction?

No. n represents a position in the list of terms — the 1st term, the 2nd term, the 10th term, and so on — so it must be a positive whole number. This calculator requires n to be a positive integer and will show an error if a decimal, fraction, zero, or negative value is entered.

Is compound interest really a geometric sequence?

Yes. Each compounding period multiplies your balance by the same factor (1 + rate), which is exactly the definition of a common ratio. The balance after n periods is a geometric sequence with a₁ equal to the starting principal and r equal to (1 + periodic interest rate) — the same math this calculator uses for a_n, just applied to money instead of an abstract list of numbers. The compound interest calculator handles the money-specific version of this formula directly.