Warm Up Find the first 5 terms of each sequence. 1. 4 . 2. 5. 3.

1. Warm Up Find the first 5 terms of each sequence. 1.4. 2. 5. 3.

2. Arithmetic Sequences Objectives: Find the indicated terms of an arithmetic sequence. Write recursive and explicit rules for arithmetic sequences.

3. The cost of mailing a letter in 2005 gives the sequence 0.37, 0.60, 0.83, 1.06, …. This sequence is called an arithmetic sequencebecause its successive terms differ by the same number d (d ≠ 0), called the common difference. For the mail costs, d is 0.23, as shown.

4. Recall that linear functions have a constant first difference. Notice also that when you graph the ordered pairs (n, an) of an arithmetic sequence, the points lie on a straight line. Thus, you can think of an arithmetic sequence as a linear function with sequential natural numbers as the domain.

5. Example 1A: Identifying Arithmetic Sequences Determine whether the sequence could be arithmetic. If so, find the common first difference and the next term. –10, –4, 2, 8, 14, … –10, –4, 2, 8, 14 Differences6 6 6 6 The sequence could be arithmetic with a common difference of 6. The next term is 14 + 6 = 20.

6. Example 1B: Identifying Arithmetic Sequences Determine whether the sequence could be arithmetic. If so, find the common first difference and the next term. –2, –5, –11, –20, –32, … –2, –5, –11, –20, –32 Differences –3 –6 –9 –12 The sequence is not arithmetic because the first differences are not common.

7. Differences –0.7 –0.7 –0.7 –0.7 Check It Out! Example 1a Determine whether the sequence could be arithmetic. If so, find the common difference and the next term. 1.9, 1.2, 0.5, –0.2, –0.9, ... 1.9, 1.2, 0.5, –0.2, –0.9 The sequence could be arithmetic with a common difference of –0.7. The next term would be –0.9 – 0.7 = –1.6.

8. Check It Out! Example 1b Determine whether the sequence could be arithmetic. If so, find the common difference and the next term. Differences The sequence is not arithmetic because the first differences are not common.

9. Each term in an arithmetic sequence is the sum of the previous term and the common difference. This gives the Recursive Rule for Arithmetic Sequences: an= an – 1 + d

10. Notice the pattern in the table. Each term is the sum of the first term and a multiple of the common difference. This pattern can be generalized into a rule for all arithmetic sequences.

11. Explicit Rule for Arithmetic Sequences:

12. Example 2: Finding the nth Term Given an Arithmetic Sequence Find the 12th term of the arithmetic sequence 20, 14, 8, 2, 4, .... Step 1 Find the common difference: d = 14 – 20 = –6.

13. Example 2 Continued Step 2 Evaluate by using the formula. an =a1 + (n – 1)d Explicit rule. Substitute 20 for a1, 12 for n, and –6 for d. a12 = 20 + (12 – 1)(–6) = –46 The 12th term is –46. Check Continue the sequence. 

14. Check It Out! Example 2a Find the 11th term of the arithmetic sequence. –3, –5, –7, –9, … Step 1 Find the common difference: d = –5 – (–3)= –2. Step 2 Evaluate by using the formula. an =a1 + (n – 1)d Explicit rule. Substitute –3 for a1, 11 for n, and –2 for d. a11= –3 + (11 – 1)(–2) = –23 The 11th term is –23.

15. Check It Out! Example 2a Continued Check Continue the sequence.  –3 –5 –7 –9 –11 –13 –15 –17 –19 –21 –23

16. Find the missing terms in the arithmetic sequence 17, , , , –7. Example 3: Finding Missing Terms Step 1 Find the common difference. an =a1 + (n – 1)d Explicit rule. Substitute –7 for an, 17 for a1, and 5 for n. –7 = 17 + (5 – 1)(d) –6 = d Solve for d.

17. Example 3 Continued Step 2 Find the missing terms using d= –6 and a1 = 17. a2= 17 + (2 – 1)(–6) = 11 The missing terms are 11, 5, and –1. a3 = 17 +(3 – 1)(–6) = 5 a4 = 17 + (4 – 1)(–6) = –1

18. 2, , , , 0. Check It Out! Example 3 Find the missing terms in the arithmetic sequence Step 1 Find the common difference. an= a1 + (n – 1)d General rule. 0= 2 + (5 – 1)d Substitute 0 for an, 2 for a1, and 5 for n. –2= 4d Solve for d.

19. Step 2 Find the missing terms using d= and a1= 2. The missing terms are Check It Out! Example 3 Continued =1