20.2. Arithmetic and Harmonic Progressions

20.2.1. Recursion Formula

We stated earlier that an arithmetic progression (or AP) is a sequence of terms in which each term after the first equals the sum of the preceding term and a constant, called the common difference, d. If an is any term of an AP, the recursion formula for an AP is as follows:


Each term of an AP after the first equals the sum of the preceding term and the common difference.

Example 12:

The following sequences are arithmetic progressions. The common difference for each is given in parentheses:

  1. 1, 5, 9, 13, ... (d = 4)

  2. 20, 30, 40, 50, ... (d = 10)

  3. 75, 70, 65, 60, ... (d = −5)

We see that each series is increasing when d is positive and decreasing when d is negative.

20.2.2. General Term

For an AP whose first term is a and whose common difference is d, the terms are

a, a + d, a + 2d, a + 3d, a + 4d, ...

We see that each term is the sum of the first term and a multiple of d, where the coefficient of d is one less than the number n of the term. So the nth term an is given by the following equation:


The nth term of an AP is found by adding the first term and (n − 1) times the common difference.

Example 13:

(a) Write the general term of an AP whose first term is 5 and has a common difference of 4. (b) Find ...

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