# Arithmetic Progressions

An arithmetic progression is a sequence in which the difference between any two consecutive terms is the same. This is the same as saying: each term exceeds the previous term by a fixed amount.
For example, 0, 6, 12, 18, . . . is an arithmetic progression in which the common difference is 6. The sequence 8, 4, 0, –4, . . . is arithmetic with a common difference of –4.

Example

The seventh number in a sequence of numbers is 31 and each number after the first number in the sequence is 4 less than the number immediately preceding it. What is the fourth number in the sequence?

1. 15
2. 19
3. 35
4. 43
5. 51
Solution

Since each number “in the sequence is 4 less than the number immediately preceding it,” the sixth term is 31 + 4 = 35; the fifth number in the sequence is 35 + 4 = 39; and the fourth number in the sequence is 39 + 4 = 43.

Following is the sequence written out:

55, 51, 47, 43, 39, 35, 31, 27, 23, 19, 15, 11, . . .

Students with strong backgrounds in mathematics may prefer to solve sequence problems by using formulas.

Note, none of the formulas in this section are necessary to answer questions about sequences on the SAT.

Since each term of an arithmetic progression “exceeds the previous term by a fixed amount,” we get the following:

 first term a + 0d where a is the first term and d is the common difference second term a + 1d third term a + 2d fourth term a + 3d . . . nth term a + (n – 1)d This formula generates the nth term

The sum of the first n terms of an arithmetic sequence is