NumberTally Sequence Calculator
Find the nth term and the sum of the first n terms of arithmetic and geometric sequences. Enter the first term, the common difference or ratio, and n — get the full term table, a growth plot, and an arithmetic-vs-geometric comparison. Results update instantly. Free, no sign-up.
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How Sequences Work
An arithmetic sequence adds a fixed difference each step: 2, 5, 8, 11, … The nth term is a₁ + (n−1)d and the sum of the first n terms is n/2 × (2a₁ + (n−1)d) — Gauss's famous trick of pairing first and last terms.
A geometric sequence multiplies by a fixed ratio: 2, 6, 18, 54, … The nth term is a₁ × r^(n−1) and the sum is a₁(r^n − 1)/(r − 1). Geometric growth starts slow and then explodes — the plot below makes the difference visceral.
Sequences in the Real World
Loan balances, depreciation schedules, and clock arithmetic are arithmetic sequences; compound interest, viral spread, and Moore's law are geometric. The classic legend: a chessboard's grains of rice doubling per square (geometric, r=2) totals more rice than has ever been grown.
Frequently Asked Questions
What is the 5th term of 2, 5, 8, …?
14. It's arithmetic with a₁=2, d=3: a₅ = 2 + 4×3 = 14, and the sum of the first 5 terms is 40.
How do you find the sum of a geometric series?
S = a₁(r^n − 1)/(r − 1). For 2, 6, 18 with n=3: 2×(27−1)/2 = 26.
What is the difference between arithmetic and geometric?
Arithmetic adds a constant difference (linear growth); geometric multiplies by a constant ratio (exponential growth).
What happens when r = 1 in a geometric sequence?
Every term equals a₁, and the sum is just n × a₁. The formula needs this special case (division by r−1 = 0).
Can the ratio be negative or fractional?
Yes. A negative ratio alternates signs (2, −6, 18, …); a ratio between −1 and 1 shrinks toward zero.