By · Updated

Arithmetic Sequence Calculator

Nth Term, Common Difference and Sum of the First n Terms, With Working

No ratings yet
Result

—

OneTask is free, with no sign-up, and built by one person.
If this tool saved you time, a coffee helps keep new ones coming.

How the Arithmetic Sequence Calculator works

This arithmetic sequence calculator finds the nth term, the sum of the first n terms and the rule for any term, written as aₙ = 4n − 1 or similar. Give it the first term and the common difference, or leave one of those blank and enter any other term instead, and it works out the missing value. Every line of working is shown, and the first 24 terms are listed with running totals.

An arithmetic sequence goes up or down by the same amount each time. 3, 7, 11, 15 adds 4 at every step, so its common difference is 4. Because the step never changes, any term can be reached directly from the first one without listing everything in between, and the whole sum has a short formula too.

How the arithmetic sequence calculator works

The nth term formula is aₙ = a₁ + (n − 1)d: start at the first term and add the difference n − 1 times. The sum of the first n terms is Sₙ = n(a₁ + aₙ) ÷ 2, the number of terms times the average of the first and last. If you give another term aₘ instead of d, the calculator finds d = (aₘ − a₁) ÷ (m − 1); if you give aₘ instead of a₁, it finds a₁ = aₘ − (m − 1)d.

First example. For 3, 7, 11, … the 10th term is 3 + 9 × 4 = 39 and the first ten terms add up to 10 × (3 + 39) ÷ 2 = 210. The rule is aₙ = 4n − 1.

Second example. A sequence starts at 5 and its 8th term is 40. The difference is (40 − 5) ÷ 7 = 5, so the 20th term is 5 + 19 × 5 = 100.

How many seats are in a theatre with 20 rows, 12 seats in the front row and 2 more in each row behind? The last row has 12 + 19 × 2 = 50 seats, so the total is 20 × (12 + 50) ÷ 2 = 620.

The first 24 terms of 3, 7, 11, … with running totals

n Term aₙ Sum Sₙ
1 3 3
2 7 10
3 11 21
4 15 36
5 19 55
6 23 78
7 27 105
8 31 136
9 35 171
10 39 210
11 43 253
12 47 300
13 51 351
14 55 406
15 59 465
16 63 528
17 67 595
18 71 666
19 75 741
20 79 820
21 83 903
22 87 990
23 91 1,081
24 95 1,176

Tips

FAQ

How do you find the common difference?

Subtract any term from the one after it. In 4, 9, 14, 19 the difference is 9 − 4 = 5. If the terms you know are further apart, divide the change by the number of steps: from a₂ = 9 to a₆ = 29 is 20 over 4 steps, so d = 5.

What is the difference between a sequence and a series?

A sequence is the list of terms, such as 2, 4, 6, 8. A series is what you get when you add them, 2 + 4 + 6 + 8 = 20. The calculator gives both: the term you ask for and the sum up to it.

How do you add the numbers from 1 to 100?

Pair them from the ends: 1 + 100, 2 + 99 and so on make 50 pairs of 101, which is 5,050. That is the sum formula with a₁ = 1, aₙ = 100 and n = 100. Enter 1, 1 and 100 to check it.

How do I know if a number is in the sequence?

Rearrange the rule for n. For aₙ = 4n − 1, the value 99 would need 4n = 100, so n = 25 and 99 is the 25th term. If n comes out as a fraction, the number is not in the sequence.

See also: all Trigonometry and Calculus tools, Arithmetic, Number Theory, Algebra and Geometry, Statistics.

Related tools

Geometric Sequence Calculator

The nth term, the sum and the infinite sum, with the working.

Summation Calculator

Sigma notation sums with the terms written out.

Linear Equation Calculator

Solve one-variable linear equations with full working.

Function Grapher

Graph up to three functions with roots and turning points marked.

Definite Integral Calculator

The integral between two limits, the area and the average value.