SeqLab

Series & Sequences Calculator

Sequences and their partial sums.

First term a₁

Common difference d

Term index n

aₙ (nth term)

aₙ = a₁ + (n−1)d

Sₙ (sum of n terms): 155

What you entered

First term a₁: 2Common difference d: 3Term index n: 10

1
nth term: a₁ + (n−1)d
2 + (10−1) × 3= 29
2
Sum of first n terms
(10/2) × (2×2 + (10−1)×3)= 155

Result

aₙ (nth term): 29

Starting at 2 with common difference 3, term 10 is 29, and the first 10 terms sum to 155.

Arithmetic series: nth term and running total

A series is the sum of a sequence's terms. These rows show both the nth term and the total of all terms up to it.

First term a1	Common difference d	Terms n	nth term	Sum of n terms
2	3	10	29	155
5	5	20	100	1050
1	2	100	199	10000
10	-2	8	-4	24
0	7	15	98	735

The third row is the classic result that the first 100 odd numbers sum to exactly 10,000 - a perfect square, which is true for any count of consecutive odd numbers starting at 1. Row four runs backwards with a negative difference, so the tenth term is below zero while the sum stays positive. The sum formula is n times the average of the first and last terms, which is why it is so much faster than adding the terms one by one: Gauss reportedly used it as a schoolboy to add 1 to 100 in seconds.

A sequence vs. a series

A sequence is the list of terms itself (2, 5, 8, 11…); a series is what you get when you add those terms together — the running total up to any point is called a partial sum.

Why partial sums matter

Total interest paid over a loan's life, total distance covered by a bouncing ball, or total seats across all rows of a stadium are all partial-sum problems in disguise.

Frequently asked questions

What's the total of 5 + 10 + 15 + … + 100?

This is an arithmetic series with a₁ = 5, d = 5. First find n: aₙ = a₁ + (n−1)d → 100 = 5 + (n−1)5 → n = 20 terms. Sum = 20(5 + 100)/2 = 1,050.

How is a series different from a sequence?

A sequence is the ordered list of numbers (2, 4, 6, 8…). A series is the sum of those numbers (2 + 4 + 6 + 8 + …). Use the arithmetic sequence calculator or geometric sequence calculator if you only need individual terms, not the running total.

Can an infinite series have a finite sum?

Yes — if the terms shrink fast enough. A geometric series with |r| < 1 converges to a₁/(1 − r). For example, 1 + ½ + ¼ + ⅛ + … = 1/(1 − 0.5) = 2. But an arithmetic series with d ≠ 0 always diverges because terms don't shrink.

I need the sum of the first 50 terms of 3, 7, 11, 15… — is there a shortcut?

Yes: it's arithmetic with a₁ = 3 and d = 4. The 50th term is 3 + 49 × 4 = 199. Sum = 50 × (3 + 199) / 2 = 5,050. No need to add all 50 terms individually.