Arithmetic Sequence Calculator

Find the nth term, sum, and list of terms for any arithmetic sequence with step-by-step solutions.

Enter Sequence Parameters

Sequence: 3, 8, 13, 18, 23, 28, 33, 38, 43, 48

Result

nth Term (an)
48
10th term
First Term (a1) 3
Common Difference (d) 5
Number of Terms (n) 10
nth Term (a_n) 48
Sum of n Terms (S_n) 255
Mean (average) 25.5

Step-by-Step Solution

a_n = a1 + (n-1)d

What Is an Arithmetic Sequence?

An arithmetic sequence (also called an arithmetic progression) is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference and is usually denoted by d. For example, the sequence 2, 5, 8, 11, 14 is arithmetic with a common difference of 3.

Arithmetic sequences are among the simplest and most important types of sequences in mathematics. They appear naturally in many contexts, from counting by a fixed increment to modeling linear growth.

Key Arithmetic Sequence Formulas

nth Term

Find any term in the sequence.

a_n = a1 + (n - 1) x d

Sum of n Terms

Add up the first n terms.

S_n = n/2 x (2a1 + (n-1)d)

Sum (alternate form)

Using first and last term.

S_n = n/2 x (a1 + a_n)

Common Difference

From any two consecutive terms.

d = a_(n+1) - a_n

Number of Terms

How many terms from a1 to a_n.

n = (a_n - a1) / d + 1

Arithmetic Mean

The average of all n terms.

Mean = (a1 + a_n) / 2

How to Find the nth Term

The nth term formula a_n = a1 + (n-1)d allows you to find any term in the sequence without listing all the previous terms. Here, a1 is the first term, d is the common difference, and n is the position of the term you want to find.

Example

Find the 20th term of the arithmetic sequence starting with 4 and common difference 7:

  1. a1 = 4, d = 7, n = 20
  2. a_20 = 4 + (20-1) x 7 = 4 + 19 x 7 = 4 + 133 = 137

How to Find the Sum

The sum of the first n terms of an arithmetic sequence can be found using: S_n = n/2 x (2a1 + (n-1)d). This formula was famously discovered by the young Carl Friedrich Gauss, who reportedly used it to quickly sum the numbers from 1 to 100 in school.

Gauss's Story

Legend has it that when Gauss was 10 years old, his teacher asked the class to add up all the numbers from 1 to 100. While the other students laboriously added one number at a time, Gauss immediately recognized that pairing the first and last terms (1+100=101, 2+99=101, etc.) gave 50 pairs of 101, for a total of 5,050.

Properties of Arithmetic Sequences

  • The graph of an arithmetic sequence forms a straight line when plotted (n vs. a_n).
  • The arithmetic mean of any two terms equals the term halfway between them.
  • If the common difference is positive, the sequence is increasing; if negative, it is decreasing.
  • If d = 0, all terms are equal (a constant sequence).
  • The sum formula gives a quadratic function of n, meaning cumulative sums grow parabolically.

Arithmetic vs. Geometric Sequences

In an arithmetic sequence, terms differ by a constant (additive). In a geometric sequence, terms differ by a constant ratio (multiplicative). For example, 2, 4, 6, 8 is arithmetic (d=2), while 2, 4, 8, 16 is geometric (ratio=2). Both are fundamental in algebra and analysis.

Practical Applications

  • Finance: Simple interest calculations, straight-line depreciation, and fixed payment schedules.
  • Construction: Evenly spaced elements like fence posts, steps, and floor tiles.
  • Physics: Uniformly accelerated motion (velocity at equal time intervals).
  • Computer science: Loop counters, memory addressing, and algorithm analysis.
  • Music: Equally tempered scales and rhythmic patterns.