1. Arithmetic Sequences Indicators: PFA#1,2 Created by Anny Lin, Crestwood Middle School Recognize arithmetic sequences Extend and write formulas for arithmetic sequences

2. A sequence is a set of numbers called terms, in a specific order. If the difference between successive terms is constant, then it is called arithmetic sequence. The difference between the terms is called the common difference.

3. Common Difference Term: Arithmetic Sequence: You can use the common difference of an arithmetic sequence to find the next term in the sequence. http://www.basic-mathematics.com/images/sequence1.gif

4. In other words, an arithmetic sequence is a numerical pattern that increases or decreases at a constant rate or value called the common difference. 3, 7, 11, 15,… 2, 6, 10, 14,… 13, 9, 5, 1,… 16, 12, 9, 8, 4, 2,… What about -17, -8, 3,14,25,…? This is Arithmetic. Arithmetic Not Arithmetic

5. Is the following sequence arithmetic? 1.Yes 2.No 0.7, 0.5, 0.3, 0.1,… The common difference is -0.2.

6. 1. 3, 6, 12, 24,… 2. -7, -3, 1, 5,... 3. 1/5, 1/7, 1/9, 1/11,... 4. -10, 5, -5/2, 5/4,... Which sequence is an arithmetic sequence?

7. 1. -7, 0, 7, 14,… 2. 0, 1/2, 1, 3/2,... 3. 10, 6, 2, -2,... 4. 2, 4, 8, 16,... Which sequence is NOT an arithmetic sequence?

8. Is the following sequence arithmetic? 1.Yes 2.No 1/2, 1/4, 1/8, 1/16,…

9. Is the following sequence arithmetic? 1.Yes 2.No

10. You can use the common difference of an arithmetic sequence to find the next term in the sequence.

11. Find the next three terms of the arithmetic sequence. 1)-15, -13, -11, -9,… What is the common difference? 2 What is the next term? -7 What are the next three terms? -7, -5, -3

12. Find the next three terms of the arithmetic sequence. 1)1/8, 1/4, 3/8,1/2,… What is the common difference? 1/8 What is the next term? 5/8 What are the next three terms? 5/8, ¾, 7/8

13. Real-World Example BOOKS The arithmetic sequence 36, 39, 42, 45, … represents the number of books in Jacob’s collection at the end of each month. Find the next three terms. Find the common difference by subtracting successive terms. The common difference is +3. Add 3 to the last term of the sequence to get the next term in the sequence. Continue adding 3 until the next three terms are found.

14. P 168 #1-4, 12-19 all Worksheet for third period

16. Words: Each term of an arithmetic sequence after the first term can be found by adding the common difference to the preceding term. Writing Arithmetic Sequence

17. Symbols: An arithmetic sequence, a 1, a 2,…, can be found as following: a 1, a 2 =a 1 +d, a 3 =a 2 +d, a 4 = a 3 +d…, Where d is the common differences a 1 is the first term, a 2 is the second term, and so on.

18. Given the arithmetic sequence 3, 6, 9, 12, 15,… Find a 1, a 2, a 3, a 4. a 1 =3 a 2 =6 a 3 =9 a 4 =12 What is a 5 ? a 5 =15 What about a 6 ? a 6 =18 What is the common difference? 3

19. Given the arithmetic sequence 14, 28, 42, 56,… Find a 1, a 2, a 3, a 4. a 1 =14 a 2 =28 a 3 =42 a 4 =56 What is a 5 ? a 5 =70 What about a 6 ? a 6 =84 What is the common difference? 14

20. a 1 =6 a 4 = a 3 +d a 2 =a 1 +d =6+9 =15 =15+9 =24 a 3 =a 2 +d =24+9 =33 Given a 1 =6 and common differences: d=9. What is a 2 ?

21. a 1 =24 common differences: d=-8 For example: a 4 = a 3 +d a 2 =a 1 +d =24-8 =16 =16-8 =8 a 3 =a 2 +d =8-8 =0

22. 25, 23, 21, 19, …, What is a 1 ? What is a 2 ? 25 23 What is the common difference d? d=-2 a 2 =a 1 +d =25+(-2) =23

23. 25, 23, 21, 19, …, What is a 3 ? =21 d=-2 a 3 =a 2 +d = a 1 +d+d a 3 = a 1 +2d =25+2(-2) =25-4 What is a 4 ? a 4 = a 3 +d = a 1 +d+d+d a 4 = a 1 +3d =25+3(-2) =25-6 =19

24. Watch Out!!! Do you see the pattern??

25. What is the relationship between a 4 and 3d? What is the relationship between a 3 and 2d?

26. What is the relationship between a n and (n-1)d?

27. 25, 27, 29, 31, …, d=2 a n =a 1 +( )d = a 1 +(n-1)d =25 +(n-1)(2) What is a n ? What is a 4 ? a 4 = a 3 + d = a 2 + d+d = a 1 + d+d+d a 4 = a 1 +3d =25+3(2) =25 + 6 =31

28. Use the smart board

29. TermSymbolIn terms of a 1 and d numbers First terma1a1 a1a1 5 Second terma2a2 a 1 +d5+1(2)=7 Third terma3a3 a 2 +d =(a 1 +d)+d =a 1 +2d 5+2(2)=9 Fourth terma4a4 a 3 +d =(a 1 +2d)+d =a 1 +3d 5+3(2)=11 :::: nth termanan a n +d =a 1 +(n-1)d 5+(n-1)(2)

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