1. Splash Screen

2. P. 692
Concept

3. an + 1 = 2an + 7 Recursive formula a1 + 1 = 2a1 + 7 n = 1
Use a Recursive Formula
Find the first five terms of the sequence in which a1 = 5 and an + 1 = 2an + 7, n ≥ 1.
an + 1 = 2an + 7 Recursive formula
a1 + 1 = 2a1 + 7 n = 1
a2 = 2(5) + 7 or 17 a1 = 5
a2 + 1 = 2a2 + 7 n = 2
a3 = 2(17) + 7 or 41 a2 = 17
a3 + 1 = 2a3 + 7 n = 3
a4 = 2(41) + 7 or 89 a3 = 41
Example 1

4. Answer: The first five terms of the sequence are 5, 17, 41, 89, 185.
Use a Recursive Formula
a4 + 1 = 2a4 + 7 n = 4
a5 = 2(89) + 7 or 185 a4 = 89
Answer: The first five terms of the sequence are 5, 17, 41, 89, 185.
Example 1

5. Find the first five terms of the sequence in which a1 = 2 and an + 1 = 3an + 2, n ≥ 1.
B. 2, 8, 26, 80, 242
C. 2, 6, 20, 62, 188
D. 26, 80, 242, 728, 2186
Example 1

6. A. Write a recursive formula for the sequence. 3, 10, 17, 24, 31, …
Write Recursive Formulas
A. Write a recursive formula for the sequence. 3, 10, 17, 24, 31, …
Step 1 Determine whether the sequence is arithmetic or geometric.
The sequence is arithmetic because each term after the first can be found by adding the common difference.
Step 2 Find the common difference.
d = 10 – 3 or 7
Example 2A

7. Step 3 Write a recursive formula.
Write Recursive Formulas
Step 3 Write a recursive formula.
an = an–1 + d Recursive formula for arithmetic sequence
an = an–1 + 7 d = 7
Answer: A recursive formula for the sequence is an = an–1 + 7, where a1 = 3.
Example 2A

8. B. Write a recursive formula for the sequence. 5, 20, 80, 320, 1280, …
Write Recursive Formulas
B. Write a recursive formula for the sequence. 5, 20, 80, 320, 1280, …
Step 1 Determine whether the sequence is arithmetic or geometric.
The sequence is geometric because each term after the first can be found after multiplying by the common ratio.
Step 2 Find the common ratio.
Example 2B

9. Step 3 Write a recursive formula.
Write Recursive Formulas
Step 3 Write a recursive formula.
an = r ● an–1 Recursive formula for geometric sequence
an = 4 ● an–1 r = 4
Answer: A recursive formula for the sequence is an = 4 ● an–1, where a1 = 5.
Example 2B

10. C. Write a recursive formula for the sequence. a3 = 6 and d = 5
Write Recursive Formulas
C. Write a recursive formula for the sequence. a3 = 6 and d = 5
Step 1 Determine whether the sequence is arithmetic or geometric.
Because d is given, the sequence is arithmetic.
Step 2 Write a recursive formula.
an = an–1 + d Recursive formula for arithmetic sequence
an = an–1 + 5 d = 5
Answer: A recursive formula for the sequence is an = an–1 + 5, where a1 = –4.
Example 2C

11. A. Write a recursive formula for the sequence. 6, 17, 28, 39, 50, …
A. a1 = 6; an = an–1 + 11
B. a1 = 6; an = 6 ● an–1 + 11
C. a1 = 6; an = an–1 + 6
D. a1 = 6; an = 11 ● an–1
Example 2

12. B. Write a recursive formula for the sequence. 4, 18, 81, 364. 5, 1640
A. a1 = 4; an = an–
B. a1 = 4; an = 4 ● an–1 + 14
C. a1 = 4; an = an–1 + 14
D. a1 = 4; an = 4.5 ● an–1
Example 2

13. C. Write a recursive formula for the sequence. a4 = 216 and r = 3
A. a1 = 8; an = an–1 + 3
B. a1 = 24; an = 3an–1
C. a1 = 8; an = 3an–1
D. a1 = 24; an = 8an–1
Example 2

14. x1 = f(x0) Iterate the function. = f(5) x0 = 5
Iterate a Function
Find the first three iterates x1, x2, and x3 of the function f(x) = 3x – 1 for an initial value of x0 = 5.
To find the first iterate x1, find the value of the function when x0 = 5.
x1 = f(x0) Iterate the function.
= f(5) x0 = 5
= 3(5) – 1 or 14 Simplify.
To find the second iterate x2, substitute x1 for x.
x2 = f(x1) Iterate the function.
= f(14) x1 = 14
= 3(14) – 1 or 41 Simplify.
Example 4

15. Substitute x2 for x to find the third iterate.
Iterate a Function
Substitute x2 for x to find the third iterate.
x3 = f(x2) Iterate the function.
= f(41) x2 = 41
= 3(41) – 1 or 122 Simplify.
Answer: The first three iterates are 14, 41, and 122.
Example 4

16. Find the first three iterates x1, x2, and x3 of the function f(x) = 2x + 1 for an initial value of x0 = 2.
A. 3, 5, 11
B. 2, 5, 11
C. 5, 11, 23
D. 11, 23, 47
Example 4

17. Homework P. 695 # 12 – 42 (x3)

18. End of the Lesson