Number Sequence Generator & Solver

What this tool is used for

• Producing an arithmetic or geometric sequence for a lesson.
• Generating terms to check a formula against.
• Producing a series for a puzzle or a test fixture.
• Seeing how quickly a geometric ratio grows.
• Generating a sequence long enough to plot its growth curve.

Frequently Asked Questions

When does an infinite geometric series have a finite sum?

Only when the common ratio is between −1 and 1, in which case the terms shrink fast enough that the total converges on a₁/(1 − r). At exactly 1 the terms never shrink and at −1 they oscillate forever; beyond that the series grows without bound and no sum exists.

What does the fractions option do differently?

It keeps exact rational arithmetic instead of floating point. A step of 1/3 in decimal accumulates rounding error every term, so a sequence that should reach exactly 10 lands on 9.999999999999998. As fractions, each term is a reduced numerator over denominator and stays exact.

Why does the nth term lookup exist when I can generate the list?

Because the closed forms — a₁ + (n − 1)d and a₁ × r^(n − 1) — reach any term directly. The 100,000th term arrives instantly without building the 99,999 before it, and the generator is capped at 200 terms precisely because listing is the expensive way to answer that question.

Why is the solved value for n shown as both rounded and exact?

Because a term number must be a whole number, but the arithmetic rarely produces one exactly. A raw result of 7.9998 means the target is term 8 with a little floating-point drift; a raw 7.4 means the value you gave is not in the sequence at all. Showing only the rounded figure would hide that difference.

Can it recognise Fibonacci or square numbers?

No. Those are recursive or polynomial rules, whereas this handles the two sequences defined by a constant step — added for arithmetic, multiplied for geometric. Fibonacci needs each term from the previous two, so it has its own generator rather than a common difference or ratio.

Common errors and gotchas

• Confusing a common difference with a common ratio, which produce very different sequences.
• Generating a geometric sequence with a large ratio and overflowing quickly.
• Disagreeing on whether the first term is index zero or one.
• Assuming a sequence continues indefinitely in a system with fixed-width numbers.
• Treating a finite sample as evidence of the general rule.