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.
The following sequences are arithmetic progressions. The common difference for each is given in parentheses:
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.
(a) Write the general term of an AP whose first term is 5 and has a common difference of 4. (b) Find ...