Sequence & Series Calculator

Calculate nth term and sum of arithmetic or geometric sequences.

Arithmetic Sequence

Constant difference

Geometric Sequence

Constant ratio

nth Term (aₙ)

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Sum (Sₙ)

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Quick Examples

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

aₙ = a₁ + (n−1)d

Sₙ = n(a₁ + aₙ)/2 = n[2a₁ + (n−1)d]/2

Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms.

aₙ = a₁ · r^(n−1)

Sₙ = a₁(rⁿ − 1)/(r − 1) for r ≠ 1

Examples

Type	First terms	10th term	Sum of 10
Arithmetic (d=3)	2, 5, 8, 11, ...	29	155
Geometric (r=2)	2, 4, 8, 16, ...	1024	2046

What Is a Sequence?

A sequence is an ordered list of numbers where each number is called a term. The position of a term is written as a subscript: a₁ is the first term, a₂ the second, and a_n the nth term. The rule that generates the terms is what distinguishes one sequence type from another. A series is what you get when you add the terms of a sequence together.

How to Identify the Sequence Type

Given a few terms, check the gaps first, then the ratios:

Test	Result	Type	Example
Subtract consecutive terms	Same difference every time	Arithmetic	3, 7, 11, 15 (d = 4)
Divide consecutive terms	Same ratio every time	Geometric	2, 6, 18, 54 (r = 3)
Neither is constant	No fixed d or r	Neither	2, 3, 5, 9 (powers of 2 plus 1)

A sequence is arithmetic if the differences repeat; it is geometric if the ratios repeat.

Arithmetic Sequences in Depth

Each term adds a fixed common difference d to the previous one. The nth term and the sum of the first n terms are:

a_n = a₁ + (n − 1)d S_n = n(a₁ + a_n) / 2 = n[2a₁ + (n − 1)d] / 2

Worked example: For 3, 7, 11, 15, … (a₁ = 3, d = 4): the 10th term is a₁₀ = 3 + 9(4) = 39, and the sum of the first 10 terms is S₁₀ = 10(6 + 36)/2 = 210. Check by pairing: (3 + 39) + (7 + 35) + … = 10 pairs × 21 = 210 ✓

The sum formula is the famous trick attributed to the young Gauss: pair the first and last terms (a₁ + a_n), note every pair sums to the same value, and multiply by the number of pairs, n/2.

Geometric Sequences in Depth

Each term multiplies the previous one by a fixed common ratio r. The formulas are:

a_n = a₁ · rⁿ⁻¹ S_n = a₁(rⁿ − 1) / (r − 1) (r ≠ 1)

Worked example: For 2, 6, 18, 54, … (a₁ = 2, r = 3): the 5th term is a₅ = 2 · 3⁴ = 162, and the sum of the first 5 terms is S₅ = 2(3⁵ − 1)/2 = 242. Check: 2 + 6 + 18 + 54 + 162 = 242 ✓

Infinite Geometric Series

When |r| < 1, the terms shrink toward zero, and the infinite sum converges to a finite value:

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

Worked example: 8 + 4 + 2 + 1 + … has a₁ = 8 and r = 1/2, so S_∞ = 8/(1 − 1/2) = 16. Check with partial sums: after 4 terms the sum is 15, after 6 terms 15.75 — clearly approaching 16. When |r| ≥ 1 the series does not converge (the sum grows without bound or oscillates).

Arithmetic vs. Geometric: Side by Side

Feature	Arithmetic	Geometric
Pattern	Add d each step	Multiply by r each step
nth term	a₁ + (n − 1)d	a₁ · rⁿ⁻¹
Sum of n terms	n(a₁ + a_n)/2	a₁(rⁿ − 1)/(r − 1)
Growth style	Linear, steady	Exponential, accelerating
Graph of terms	Points on a straight line	Points on an exponential curve
Typical use	Fixed monthly savings, seating rows	Compound interest, population growth

Real-World Applications

Savings plans: Depositing a fixed amount each month is an arithmetic sequence; interest compounding on it is geometric.
Population growth: A population growing by a fixed percentage each year follows a geometric sequence.
Loan balances: A loan shrinking by fixed monthly payments is arithmetic; the interest portion behaves geometrically.
Computer science: Halving a search space each step (binary search) is a geometric sequence with r = 1/2.
Physics: A bouncing ball's rebound heights form a geometric sequence with r < 1.

Common Mistakes to Avoid

Off-by-one in the nth term: The exponent is n − 1, not n. The 1st term uses r⁰ = 1.
Using the sum formula at r = 1: It divides by zero. When r = 1, the sum is simply n · a₁.
Mixing up d and r: d is added (arithmetic), r is multiplied (geometric) — they are not interchangeable.
Applying S_∞ when |r| ≥ 1: The infinite sum formula only works when the terms shrink (|r| < 1).
Counting terms wrong: The "sum of the first n terms" includes a₁ — the 10th term is the last one included when n = 10.

Finding a Missing Difference or Ratio

You do not always know d or r up front — sometimes you know two terms and must work backwards:

Arithmetic: If a₃ = 11 and a₇ = 27, then 4 steps of size d bridge the gap: 4d = 27 − 11 = 16, so d = 4. Then a₁ = 11 − 2(4) = 3. Check: a₃ = 3 + 2(4) = 11 ✓, a₇ = 3 + 6(4) = 27 ✓
Geometric: If a₂ = 6 and a₅ = 48, then 3 multiplications by r bridge the gap: r³ = 48/6 = 8, so r = 2. Then a₁ = 6/2 = 3. Check: a₂ = 3 × 2 = 6 ✓, a₅ = 3 × 2⁴ = 48 ✓

Recursive vs. Explicit Formulas

Every sequence can be described two ways. The recursive form defines each term from the previous one (a₁ = 3, a_n = a_n−1 + 4); the explicit form gives any term directly (a_n = 3 + (n − 1)4). The calculator uses the explicit form because it jumps straight to the nth term without computing everything before it. Recursive thinking is useful for understanding, explicit formulas for calculating.

Sigma Notation

Sums are often written compactly with the Greek letter Σ (sigma). The sum of the first n terms of an arithmetic sequence becomes:

S_n = Σ_k=1ⁿ [a₁ + (k − 1)d]

Read it as: "add up a₁ + (k − 1)d for every integer k from 1 to n." For the sequence 3, 7, 11, … with n = 4: (3) + (7) + (11) + (15) = 36, matching S₄ = 4(6 + 12)/2 = 36 ✓

Frequently Asked Questions

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers. A series is the sum of the terms of a sequence. For example, 1, 2, 3, 4, ... is a sequence, while 1 + 2 + 3 + 4 + ... is a series.