Switch between arithmetic and geometric mode to find the nth term and the sum of the first n terms.

How to Use the Number Sequence Calculator

Toggle between Arithmetic and Geometric first — each mode swaps in its own three fields. Arithmetic asks for a first term and a common difference; geometric asks for a first term and a common ratio. Either way, set N to whichever term you want, and both the nth term and the running sum through that term appear below.

The two sequence types grow in fundamentally different ways. An arithmetic sequence adds the same amount at every step — linear growth, a straight line if you plotted it. A geometric sequence multiplies by the same factor every step instead — exponential growth, a curve that bends upward, or collapses toward zero if the ratio is a fraction.

Arithmetic nth term = first term + (n − 1) × common difference
Geometric nth term  = first term × common ratio^(n − 1)

Take the defaults: first term 3, common difference 5, tenth term. That's 3 + 9 × 5 = 48 — each step just tacks on another 5. Switch to geometric with first term 2, ratio 3, sixth term: 2 × 3⁵ = 486. Same idea of "keep applying the rule," a completely different shape of result.

Summing works differently too:

Arithmetic sum = n/2 × (2 × first term + (n − 1) × common difference)
Geometric sum  = first term × (ratio^n − 1) / (ratio − 1), or first term × n when ratio = 1

This is the same split that shows up in simple versus compound interest — simple interest adds a fixed amount each period (arithmetic), compound interest multiplies by a fixed factor each period (geometric), and the compounding version pulls further ahead every year that passes. Population models split the same way: linear growth adds a fixed number of people annually, exponential growth multiplies by a growth rate instead.