Number Sequence Calculator

Arithmetic vs. Geometric Sequences

A sequence is simply an ordered list of numbers following a rule. The two most common types are arithmetic, where the same fixed amount is added each time, and geometric, where each term is instead multiplied by the same fixed factor.

Arithmetic Sequences

$$a_n = a_1 + (n-1)d \qquad S_n = \frac{n}{2}\big(2a_1 + (n-1)d\big)$$

For example, with a first term of 3 and a common difference of 4, the sequence runs 3, 7, 11, 15, 19, and so on. The 10th term is 3 + (10 − 1) × 4 = 39, and the sum of the first 10 terms is (10 ÷ 2) × (2 × 3 + (10 − 1) × 4) = 5 × 42 = 210.

Geometric Sequences

$$a_n = a_1 \cdot r^{\,n-1} \qquad S_n = a_1 \cdot \frac{1-r^n}{1-r} \;\; (r \neq 1)$$

For example, with a first term of 2 and a common ratio of 3, the sequence runs 2, 6, 18, 54, 162, and so on — growing much faster than an arithmetic sequence with similar-looking numbers, since each term is multiplied rather than added to. The 8th term is 2 × 3⁷ = 4,374, and the sum of the first 8 terms is 2 × (1 − 3⁸) ÷ (1 − 3) = 6,560.

The Special Case: Common Ratio of Exactly 1

The geometric sum formula divides by (1 − r), which becomes division by zero when r equals exactly 1. But this case doesn't actually need the formula at all: when the ratio is 1, every single term equals the first term, so the sum of n terms is simply the first term multiplied by n — no formula required, just repeated addition of the same number.

Common Sequence Mistakes

Using n instead of (n − 1) in the exponent or multiplier is the single most common error in both formulas — the first term itself is already "term 1," so it needs zero differences or multiplications applied to it, not one. Mixing up which sequence type applies to a real-world problem is another frequent slip: compound interest and population growth are geometric (multiplicative), while a savings plan that adds the same fixed amount every period is arithmetic (additive) — using the wrong formula for the wrong situation gives a plausible-looking but wrong answer. Forgetting the special r = 1 case in a geometric sum, and getting a division-by-zero error or an undefined result, is also common when the formula is applied mechanically without checking for it first.

Sequence Terms You Should Know

Term — a single value in a sequence, referenced by its position (the 1st term, the 5th term, and so on).

Common Difference — the fixed amount added between consecutive terms of an arithmetic sequence.

Common Ratio — the fixed factor multiplied between consecutive terms of a geometric sequence.

Series — the sum of the terms of a sequence, as opposed to the sequence (the list of terms) itself.