How to Use the Sequence Induction Verifier
Mathematical induction proves that a statement holds for every natural number n. The closed-form formulas for arithmetic and geometric sequences come from the same inductive idea: apply a rule to the first term, n-1 times in a row. This calculator computes the nth term two different ways and lets you see with your own eyes that they agree.
The closed formula result plugs your values directly into the standard formula (aₙ = a₁ + (n-1)d for arithmetic, aₙ = a₁ × r^(n-1) for geometric). The inductive result starts at the first term and actually repeats the addition or multiplication step all the way to the nth term. Even though the two paths are completely different, they always land on the same value — which is exactly why the closed formula works.
Choose arithmetic to enter a common difference, or geometric to enter a common ratio. Term numbers from 1 to 1000 are supported. Note that for geometric sequences, a large ratio combined with a large term number can produce very big numbers.
Frequently Asked Questions
In theory they should always match. For very large term numbers, floating-point rounding can introduce a tiny difference, so this calculator checks a small tolerance rather than requiring an exact match.
Yes. A negative common difference produces a decreasing arithmetic sequence, and a negative common ratio produces a geometric sequence whose terms alternate in sign.