Arithmetic Sequences

1. Arithmetic Sequences

2. Definition of an arithmetic sequence. An arithmetic sequence is a sequence in which each term but the first is found by adding a constant, called the common difference d, to the previous term.

3. Example 1. The table below shows the cost of mailing a first class letter in 1995. Ounces 1 2 3 4 5 Cost .32 .55 .78 1.01 1.24 Find how much it costs to mail letters that weigh 6,7, and 8 oz.

4. Cost .32 .55 .78 1.01 1.24 Example 1. How much to mail letters that weigh 6,7, and 8 oz. First find the common difference. +.23 +.23 +.23 +.23 .23 or 23 cents is the common difference.

5. Example 1. The table below shows the cost of mailing a first class letter in 1995. d = .23 Ounces 1 2 3 4 5 Cost .32 .55 .78 1.01 1.24 6 8 7 Ounces 1.93 1.70 Cost 1.47

6. Example 2. Find the next four terms of the arithmetic sequence 91, 83, 75, .... 91 83 75 -8 -8 The common difference is -8.

7. Example 2. Find the next four terms of the arithmetic sequence 91, 83, 75, .... 67 59 51 43 91 83 75 -8 -8 The next four terms are

8. There is a pattern in the way the terms of an arithmetic sequence are formed. It is possible to develop a formula that expresses each term of an arithmetic sequence in terms of the first term a1 and the common difference d.

9. Let’s look at example 2. 91 83 75 67 numerical symbols a1 a2 a3 a4 an In terms of d where d = -8 a2 = 91+1(-8) a1 = 91+0(-8) a4 = 91+3(-8) a3 = 91+2(-8)

10. Let’s look at example 2. 91 83 75 67 numerical symbols a1 a2 a3 a4 an In terms of d a2 = 91+1(-8) a1 = 91+0(-8) a4 = 91+3(-8) a3 = 91+2(-8) Therefore an = 91+(n-1)(-8) Equivalently an = a1+(n-1)(-8)

11. Formula for the nth term of an arithmetic sequence. The nth term of an arithmetic sequence with first term a1 and common difference d is given by an = a1+(n-1)(d) where n is a positive integer.

12. Example 3. A radio station is giving away at least $1000.00 in a contest. For each caller who answers the question incorrectly the station adds $97.00 to the jackpot. If you are the 18th caller and the first to answer correctly how much do you win?

13. Example 3. Radio station contest. This is an arithmetic sequence with a1 = 1000 and d = 97. an = a1 + (n-1)d a18 = 1000 + (18-1)(97) a18 = 1000 + 17(97) = 2649 a18 = 1000 + 1649

14. The terms between any two nonconsecutive terms of an arithmetic sequence are called the arithmetic means. In the sequence 14, 23, 32, 41, 50, 59, 68, 77 32, 41, and 50 are the three arithmetic means between 23 and 59

15. Example 4. Find the four arithmetic means between 18 and 78. Use the nth term formula to find d. 18, ____, ____, ____, ___, 78 78 is a6 18 is a1

16. Example 4. Find the four arithmetic means between 18 and 78. Use the nth term formula to find d. 18, ____, ____, ____, ___, 78 18 is a1 78 is a6 a6 = a1 + 5(d) 12 = d 78 = 18 + 5(d)

17. Example 4. Find the four arithmetic means between 18 and 78. 18, ____, ____, ____, ___, 78 12 = d Now use d to find the terms a2 = 18 + 1(12) a4 = 18 + 3(12) a3 = 18 + 2(12) a5 = 18 + 4(12) Terms are 30, 42, 54, and 66

18. Example 5. Write an equation for the nth term of the arithmetic sequence 6, 13, 20, 27. ... In this sequence a1 = 6 and d = 7 Therefore an = 6 + (n-1)7 an = 6 + 7n - 7 an = 7n - 1