1. 13.1 Arithmetic and Geometric Sequences Explicit Formulas

2. Objectives : 1. Identify Arithmetic and Geometric sequences 2. Complete missing information associated with sequences 3. Define formulas for sequences Vocabulary : Sequence, term, Arithmetic, Geometric 13.1 Arithmetic and Geometric Sequences

3. Learning The Lingo A "sequence" an ordered list of numbers; the numbers in this ordered list are called "elements" or "terms".

4. Learning the Lingo A sequence may be named or referred to as "A" or "A n ". The terms of a sequence are usually named something like "a i " or "a n ", with the subscripted letter "i" or "n" being the "index" or counter. A n = a1a1 a2a2 a3a3 a4a4 a5a5

5. The two simplest sequences to work with are arithmetic and geometric sequences. An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. For instance:  Example One: 2, 5, 8, 11, 14,...  Example Two: 7, 3, –1, –5,... Arithmetic Sequences Add 3 to each term Subtract 4 to each term

6. Learning The Lingo The number added (or subtracted) at each stage of an arithmetic sequence is called the "common difference" d, because if you subtract (find the difference of) successive terms, you'll always get this common value.

7. Writing Arithmetic Sequences a n = a 1 + (n – 1)d

8. Example – Your Turn Find the common difference and the next term of the following sequence: 3, 11, 19, 27, 35,... The difference is always 8, so d = 8. Then the next term is 35 + 8 = 43.

9. Examples: Find a formula for a n and find the 10 th term 2,6,10,14,18,… 17,10,3,-4,-11,-18,…

10. Find the n-th term (formula) of the arithmetic sequence having a 4 = 93 and a 8 = 65. Examples: Since a 4 and a 8 are four places apart, then I know from the definition of an arithmetic sequence that a 8 = a 4 + 4d. 65 = 93 + 4d –28 = 4d –7 = d 93 = a + 3(–7) 93 + 21 = a 114 = a 65 = a + 7(–7) 65 + 49 = a 114 = a OR

11. Geometric Sequences A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. For instance:  Example One: 1, 2, 4, 8, 16,...  Example Two: 81, 27, 9, 3, 1, 1/3,... Multiply by 2 at each step Divide by 3 at each step

12. Learning The Lingo The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (find the ratio of) successive terms, you'll always get this common value.

13. Writing Geometric Sequences a n = a 1 r (n – 1)

14. Find the common ratio and the seventh term of the following sequence: 2/9, 2/3, 2, 6, 18,... The ratio is always 3, so r = 3. Then the sixth term is (18)(3) = 54 and the seventh term is (54)(3) =162 Example – Your Turn!

15. 1,3,9,27,81,… Examples: Find a formula for a n and find the 10 th term 64,-32,16,-8,4,…

16. Example Find the n-th (formula) of the geometric sequence with a 5 = 5/4 and a 12 = 160. These two terms are 12 – 5 = 7 places apart, so, from the definition of a geometric sequence, I know that a 12 =a 5 r 7 160 = (5/4)(r 7 ) 128 = r 7 2 = r 5/4 = a(2 4 ) = 16a 5/64 = a 160 = a(2 11 ) = 2048a 160/2048 =5/64= a OR

17. Homework: Textbook: p. 477 #17-21, 29, 33

18. Practice WB p. 91 # 4 No, multiply is Geometric

19. Practice WB p. 98 # 3 a n =3(1/4) (n-1) or a n =12(1/4) (n) a 8 =3/4 7 or a 8 =0.0001831

20. Practice WB p. 91 # 8 a n = 1 + (n – 1)-5 or a n = - 5n + 6 a 30 = -144

21. Practice WB p. 97 # 3 a n =1024(+/-1/4) (n-1) a n =4096(+/-1/4) (n)

22. Practice WB p. 91 # 12 a n = -81 + (n – 1) 11 or 11n + 92