Number Sequence Calculator

Arithmetic and geometric sequences — find the nth term and the sum of the first n terms

Arithmetic and geometric sequences follow two fundamentally different growth patterns — constant additive growth versus constant multiplicative growth — and finding the nth term or the sum of the first n terms requires using the correct formula for whichever type of sequence is actually involved, since applying the wrong formula produces a systematically incorrect result.

Why arithmetic and geometric sequences need different formulas

An arithmetic sequence adds the same fixed amount (the common difference) to get from one term to the next — growth is linear, and both the nth term and the sum-of-first-n-terms formulas reflect that constant additive step. A geometric sequence instead multiplies by the same fixed ratio (the common ratio) to get from one term to the next — growth is exponential, not linear, and both its nth-term and sum formulas are structurally different from the arithmetic versions specifically because they need to capture that multiplicative, compounding pattern rather than a simple additive one. Mistaking a geometric sequence for an arithmetic one (or vice versa) and applying the wrong formula produces a result that looks superficially plausible but is mathematically wrong, since the two growth patterns diverge increasingly the further out in the sequence you go.

A worked example

An arithmetic sequence and a geometric sequence that happen to start with the same first two terms will diverge sharply as more terms are added — the arithmetic sequence keeps adding the same fixed amount each step, while the geometric sequence's multiplicative growth compounds increasingly, producing a dramatically different value by, say, the tenth term even though the first two terms looked identical.

How this connects to your other math calculations

Geometric sequences share their underlying multiplicative-growth structure with exponential decay and half-life calculations — both are governed by repeated multiplication by a fixed ratio, just in growth versus decay directions respectively, which is why the underlying mathematical pattern (though not the specific formula) connects the two calculators conceptually.

Common mistakes

Assuming a sequence is arithmetic just because the first couple of terms show a roughly consistent-looking step, without actually checking whether the ratio (not the difference) between consecutive terms is what's actually constant, is a common misclassification — always verify whether consecutive terms share a constant additive difference (arithmetic) or a constant multiplicative ratio (geometric) before applying either formula, rather than assuming from a quick glance at just the first few terms.