Arithmetic Progression: Formulas & Examples

An arithmetic progression, usually shortened to AP, is a list of numbers in which every term after the first is found by adding the same fixed number to the previous term. That fixed number is called the common difference and is written d. For example, 5, 9, 13, 17, 21, … is an AP with first term a = 5 and common difference d = 4, because each step adds 4.

Arithmetic progressions matter because constant-rate change is everywhere. A taxi meter adding a fixed fare per kilometre, monthly savings rising by a fixed step, and staircase steps are all arithmetic progressions in disguise.

Key Concepts

• First term (a): the starting value of the sequence, also written T_1.
• Common difference (d): the fixed amount added each step, found by d = T_2 - T_1 = T_3 - T_2, and so on. It is the same for every neighbouring pair.
• General form: a, a + d, a + 2d, … , where the multiplier of d trails the term number by one.
• nth term: T_n = a + (n - 1)d. There are (n - 1) jumps of size d between the 1st and the nth term.
• Sum of n terms: S_n = n/2 times [2a + (n - 1)d], which also equals n/2 times (first term + last term).

Formulas

• nth term: T_n = a + (n - 1)d, where n is a positive whole number.
• Sum of first n terms: S_n = n/2 times [2a + (n - 1)d] = n/2 times (a + l), where l is the last term.
• Assumptions: n is a positive integer; d is constant.

Worked Example

Consider the AP with first term a = 3 and common difference d = 4, so the sequence runs 3, 7, 11, 15, … . Find the 20th term and the sum of the first 20 terms. The 20th term is T_20 = 3 + (20 - 1) times 4 = 3 + 76 = 79. The sum is S_20 = 20/2 times (first term + last term) = 10 times (3 + 79) = 10 times 82 = 820. The other formula agrees: 10 times [6 + 76] = 820.

Common Misconceptions

• Every increasing sequence is an AP: false. The sequence 1, 2, 4, 8, 16 keeps growing, but its differences (1, 2, 4, 8) change every step, so it is not an AP.
• d must be positive: the common difference can be zero, giving a constant sequence like 5, 5, 5, …, or negative, giving a decreasing sequence like 20, 15, 10, 5, … . Both are valid APs.
• T_n = a + nd: this counts one jump too many. From the 1st to the nth term there are only (n - 1) jumps.

Real-World Applications

• Finance: the balance of a simple-interest account at the end of each year forms an AP.
• Transport: taxi and auto fares with a fixed per-kilometre charge follow an AP.
• Design: theatre seating rows that gain the same number of seats per row are planned with APs.

An arithmetic progression is a sequence with a constant step size, described fully by its first term a and common difference d. From those two numbers, T_n = a + (n - 1)d reaches any term and S_n = n/2 times (first + last) totals them.