1. Homework Log Fri. 4/8 Lesson Rev Learning Objective:
To remember everything in Chapter 9!
Hw: W-S Review 2
Study for Test (Mon)
Virtual Lab (Sun 6pm)

2. 4/8/16 Chapter 9 Review
Algebra II

3. Learning Objective
To remember everything in Chapter 9!

4. Arithmetic Sequence & Series
SEQUENCE/Explicit nth term formula 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑 Sum/SERIES Formula 𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 ) Summation Notation
𝑛=1 # 𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎

5. Geometric Sequence & Series
SEQUENCE/Explicit nth term formula 𝑎 𝑛 = 𝑎 1 (𝑟) 𝑛−1 Sum/SERIES Formula (Finite) 𝑆 𝑛 = 𝑎 1 (1− 𝑟 𝑛 ) 1−𝑟 Sum/SERIES Formula (Infinite) 𝑆= 𝑎 1 1−𝑟
𝑟 <1 Converge
𝑟 ≥1 Diverge

6. Summation Notation Arithmetic Geometric 𝑛=1 # 𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎
𝑛=1 # 𝑎 1 (𝑟) 𝑛−1
𝑛=1 # 𝑎 1 + 𝑛−1 𝑑

7. Write the explicit formula & find the 30th term
1. −15, −10, −5, …
𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑
𝑎 𝑛 =−15+ 𝑛−1 (5)
𝑎 𝑛 =−15+5𝑛−5
𝑎 𝑛 =5𝑛−20
𝑎 30 =5 30 −20
𝑎 30 =130

8. Find the missing terms
2. The sequence is arithmetic 17, ___, ___, ___, 41
+ 6
+ 6
+ 6
+ 6 23 29 35
𝑎 1 = 17
n = 5
d = ___
𝑎 𝑛 = 41
𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑
41=17+ 5−1 (𝑑)
24=4𝑑
𝑑=6

9. Find the sum 3. n = 50 𝑎 1 =3 1 +4=7 𝑎 𝑛 = 𝑎 50 =3 50 +4=154
𝑛=1 50 (3𝑛+4) 3.
n = 50
𝑎 1 =3 1 +4=7
𝑎 𝑛 = 𝑎 50 = =154
𝑆 50 = 50 2 (7+(154))
𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 )
=4025

10. Solve 4. *** Number of terms = Top – Bottom + 1 n = 200 – 30 + 1 = 171
𝑛= (4𝑛+12) 4.
*** Number of terms = Top – Bottom + 1
n = 200 – = 171
𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 )
First term = 4(30) + 12
First term = 132
Last term = 4(200) + 12
Last term = 812
𝑆 171 = ( )
𝑆 171 =80712

11. Write the Summation Notation & Find the Sum
… + 112
n = ___
𝑎 1 =16
𝑎 𝑛 =112
d = 6
Use 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑 to find n
112 = 16 + (n – 1)(6)
112 = n - 6
112 = 6n + 10
102 = 6n
n = 17
𝑛=1 17 (6𝑛+10)
𝑆 17 =1088
𝑆 17 = 17 2 (16+112)
𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 )

12. Write the Summation Notation & Find the Sum
(-25)+(-12)+…+573
n = ___
𝑎 1 =−38
𝑎 𝑛 =573
d = 13
Use 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑 to find n
573 = (n – 1)(13)
573 = n - 13
573 = 13n -51
624 = 13n
n = 48
𝑛=1 48 (13𝑛−51)
𝑆 48 =12840
𝑆 48 = 48 2 (−38+573)
𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 )

13. Write explicit formula & find 8th term
Geo: r = − 1 2
7. -6, 3, − 3 2 ,… ↑
𝑎 1
𝑎 𝑛 = 𝑎 1 (𝑟) 𝑛−1
𝑎 𝑛 =−6 − 𝑛−1
NOT 𝑎 𝑛 = 3 𝑛−1
𝑎 8 =−6 − −1
𝑎 8 = 3 64

14. Find the missing terms
8. The sequence is geometric 4, ___, ___, ___, 64
x -2
x -2
x -2
x -2 -8 16
-32 8 16 32
x 2
x 2
x 2
x 2
𝑎 𝑛 = 𝑎 1 (𝑟) 𝑛−1
64=4 (𝑟) 5−1
16= (𝑟) 4
𝑟=±2
𝑎 1 = 4
n = 5
r = ___
𝑎 𝑛 = 64

15. Find the sum Infinite Geometric Series 9. 2−1+ 1 2 − 1 4 +… = −1 2
9. 2− − 1 4 +…
r = 𝑎 2 𝑎 1
= −1 2
𝑆= 𝑎 1 1−𝑟
𝑆= 2 1−(−1/2)
𝑟 <1
Converges
𝑆= 2 3/2
= 4 3
𝑛=1 ∞ 2 − 𝑛−1

16. Find the sum 10. 3−9+27+… Infinite Geometric Series r = 𝑎 2 𝑎 1 = −9 3
10. 3−9+27+…
Infinite Geometric Series
r = 𝑎 2 𝑎 1
= −9 3
𝑟=−3
𝑟 ≥1
Diverges
No Sum

17. Find the sum 𝑆 𝑛 = 𝑎 1 (1− 𝑟 𝑛 ) 1−𝑟 11. 𝑆 8 = −3(1− (3) 8 ) 1−(3)
𝑛=1 8 −3 (3) 𝑛−1
𝑆 𝑛 = 𝑎 1 (1− 𝑟 𝑛 ) 1−𝑟
11.
𝑆 8 = −3(1− (3) 8 ) 1−(3)
n = 8
𝑎 1 =−3
r = 3
𝑆 8 = −3(1−(6561)) −2
𝑆 8 = −3(−6560) −2
𝑆 8 =−9840

18. Arithmetic Sequence 12. 𝑎 1 =89 𝑎 7 =71 What is 𝑎 20 ?
𝑎 𝑛 =89+ 𝑛−1 (−3)
𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑
𝑎 20 =89+(20−1)(−3)
𝑎 7 = 𝑎 1 + 7−1 𝑑
𝑎 20 =32
71 = d
d = -3

19. Geometric Sequence 13. 𝑎 1 =3 𝑎 10 =59049 What is 𝑎 3 ? r = 3
𝑎 𝑛 = 𝑎 1 (𝑟) 𝑛−1
𝑎 𝑛 =3 (3) 𝑛−1
𝑎 10 = 𝑎 1 (𝑟) 10−1
𝑎 3 =3 (3) 3−1
59049=3 (𝑟) 9
𝑎 3 =3 (3) 2
19683= (𝑟) 9
𝑎 3 =27

20. Expand 14. ( 𝑦 2 −3𝑥) 4 ( 𝑦 2 ) 4 (−3𝑥) 0 + ( 𝑦 2 ) 3 (−3𝑥) 1 4 +
14. ( 𝑦 2 −3𝑥) 4
( 𝑦 2 ) 4
(−3𝑥) 0 1 +
( 𝑦 2 ) 3
(−3𝑥) 1 4 +
( 𝑦 2 ) 2 6
(−3𝑥) 2 +
( 𝑦 2 ) 1
(−3𝑥) 3 +
( 𝑦 2 ) 0
(−3𝑥) 4 4 1
𝑦 8
− 12 𝑦 6 𝑥
+ 54 𝑦 4 𝑥 2
−108 𝑦 2 𝑥 3
+ 81 𝑥 4

21. Find the 4th term 15. (3𝑥−2𝑦) 9 (3𝑥) 9 (−2𝑦) 0 + (3𝑥) 8 9 (−2𝑦) 1 +
15. (3𝑥−2𝑦) 9
(3𝑥) 9 1
(−2𝑦) 0 +
(3𝑥) 8 9
(−2𝑦) 1 + 36
(3𝑥) 7
(−2𝑦) 2 + 84
(3𝑥) 6
(−2𝑦) 3
− 𝑥 6 𝑦 3

22. Assignment: W-S Review 2 Study for Test (Mon) Virtual Lab (Sun 6pm)