Arithmetic Sequences A sequence in which each term after the first is obtained by adding a fixed number to the previous term is an arithmetic sequence (or arithmetic progression). The fixed number that is added is the common difference. The sequence is an arithmetic sequence since each term after the first is obtained by adding 4 to the previous term.

Arithmetic Sequences That is, and so on. The common difference is 4.

Arithmetic Sequences If the common difference of an arithmetic sequence is d, then by the definition of an arithmetic sequence, for every positive integer n in the domain of the sequence. Common difference d

Example 1 FINDING THE COMMON DIFFERENCE Find the common difference, d, for the arithmetic sequence Solution We find d by choosing any two adjacent terms and subtracting the first from the second. Choosing – 7 and – 5 gives Choosing – 9 and – 7 would give the same result. Be careful when subtracting a negative number.

Example 2 FINDING TERMS GIVEN a 1 AND d Find the first five terms for each arithmetic sequence. Solution a. The first term is 7, and the common difference is – 3. Start with a 1 = 7. Add d = – 3. Add – 3.

Example 2 FINDING TERMS GIVEN a 1 AND d Find the first five terms for each arithmetic sequence. Solution b. a 1 = – 12, d = 5 Start with a 1. Add d = 5.

Look At It This Way If a 1 is the first term of an arithmetic sequence and d is the common difference, then the terms of the sequence are given by and, by this pattern,

nth Term of an Arithmetic Sequence In an arithmetic sequence with first term a 1 and common difference d, the nth term, is given by

Example 3 FINDING TERMS OF AN ARITHMETIC SEQUENCE Find a 13 and a n for the arithmetic sequence – 3, 1, 5, 9, … Solution Here a 1 = – 3 and d = 1 – (– 3) = 4. To find a 13 substitute 13 for n in the formula for the nth term. n = 13 Work inside parentheses first. Let a 1 = – 3, d = 4. Simplify.

Example 3 FINDING TERMS OF AN ARITHMETIC SEQUENCE Find a 13 and a n for the arithmetic sequence – 3, 1, 5, 9, … Solution Find a n by substituting values for a 1 and d in the formula for a n. Let a 1 = – 3, d = 4. Simplify. Distributive property

Arithmetic Series The total amount of interest paid is given by the sum of the terms of this sequence. Now we develop a formula to find this sum without adding all 30 numbers directly. Since the sequence is arithmetic, we can write the sum of the first n terms as

Sum of the First n Terms of an Arithmetic Sequence If an arithmetic sequence has first term and common difference d, then the sum of the first n terms is given by or

Example 7 USING THE SUM FORMULAS Evaluate S 12 for the arithmetic sequence – 9, – 5, – 1, 3, 7, …. a. Solution We want the sum of the first 12 terms. Using a 1 = – 9, n = 12, and d = 4 in the second formula, gives

Example 7 USING THE SUM FORMULAS b. Solution Use a formula for S n to evaluate the sum of the first 60 positive integers. The first 60 positive integers form the arithmetic sequence 1, 2, 3, 4, …, 60. Thus, n = 60, a 1 = 1, and a 60 = 60, so we use the first formula in the preceding box to find the sum.