An arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed number, called the common difference, to the preceding term. The common difference is usually represented by the letter “d” and the first term in the sequence is usually represented by “a1”.
The nth term of an arithmetic progression is given by the formula: an = a1 + (n – 1)d
For example, consider the following arithmetic progression: 3, 7, 11, 15, 19, …
In this arithmetic progression, the common difference is 4, since each term is obtained by adding 4 to the preceding term. The first term is 3. Using the formula above, we can find the 10th term: a10 = 3 + (10 – 1)4 = 3 + 36 = 39
Arithmetic progressions are useful in solving problems in which a series of quantities is increasing or decreasing by a fixed amount.
Here are 5 examples of arithmetic progressions:
Example 1: Consider the following arithmetic progression: 2, 4, 6, 8, 10, … In this arithmetic progression, the common difference is 2 and the first term is 2. Using the formula above, we can find the 8th term: a8 = 2 + (8 – 1)2 = 2 + 14 = 16
Example 2: Consider the following arithmetic progression: -3, 1, 5, 9, 13, … In this arithmetic progression, the common difference is 4 and the first term is -3. Using the formula above, we can find the 6th term: a6 = -3 + (6 – 1)4 = -3 + 20 = 17
Example 3: Consider the following arithmetic progression: 10, 6, 2, -2, -6, … In this arithmetic progression, the common difference is -4 and the first term is 10. Using the formula above, we can find the 4th term: a4 = 10 + (4 – 1)(-4) = 10 – 12 = -2
Example 4: Consider the following arithmetic progression: 1, -1, -3, -5, -7, … In this arithmetic progression, the common difference is -2 and the first term is 1. Using the formula above, we can find the 6th term: a6 = 1 + (6 – 1)(-2) = 1 – 10 = -9
Example 5: Consider the following arithmetic progression: 8, 12, 16, 20, 24, … In this arithmetic progression, the common difference is 4 and the first term is 8. Using the formula above, we can find the 3rd term: a3 = 8 + (3 – 1)4 = 8 + 8 = 16
Quiz:
- What is an arithmetic progression?
- What is the common difference in an arithmetic progression?
- What is the formula for finding the nth term in an arithmetic progression?
- How do you find the 10th term in the arithmetic progression 3, 7, 11, 15, 19, …?
- What is the common difference in the arithmetic progression 10, 6, 2, -2, -6, …?
- What is the 3rd term in the arithmetic progression 1, -1, -3, -5, -7, …?
- How do you find the 8th term in the arithmetic progression 2, 4, 6, 8, 10, …?
- What is the common difference in the arithmetic progression 8, 12, 16, 20, 24,…?
