Post on 29-Jul-2020
Chapter 8 Sequences and Series 8.1 Arithmetic Sequences Examples of Sequences: Naming Sequences and Terms: Sequences are Functions:
Recursive Formulas for Sequences: Arithmetic (Linear) Sequences: Ex 1: Identify π1, the common difference π and an explicit formula for each sequence
a. 1, 4, 7, 10, 13, 16, β¦
b. 15, 10, 5, 0, β5, β10, β¦
c. 1,1
2, 0, β
1
2, β1, β¦
Ex 2: Find the common difference whose for the arithmetic sequence whose formula is ππ = 6π + 3 Ex 3: Find the 81st term of the sequence 3, 5, 7, 9, β¦ Ex 4: Suppose π is an arithmetic sequence with π2 = 14 and π6 = 4. Find
a. An explicit formula for π b. A recursive formula for π
Ex 5: Suppose π is an arithmetic sequence with π3 = β9 and π10 = β37
c. An explicit formula for π d. A recursive formula for π
8.2 Geometric and Other Sequences Geometric Sequences: Ex 1: Write the first 5 terms of the geometric sequence in which π1 = 4 and π = 5 Ex 2: Write the first 5 terms of the geometric sequence in which ππ = 4ππβ1 and π1 = 3. Ex 3: Use the formula for the nth term of a geometric sequence to find the 6th term of the sequence with π1 =
15625 and π =1
5
Ex 4: If you invest $10,000 in a 5-year CD with an annual yield of 4.83%, the amount you begin with and the amounts you have at the end of consecutive years form a finite geometric sequence.
a. Find an explicit formula for this sequence
b. Find a recursive formula for this sequence Ex 5: For each sequence below
a. Determine whether it could be arithmetic, geometric, or neither b. Give the next two terms of the sequence c. State an explicit formula for the sequence
1. ππ = 18, 24, 30, 36, β¦ 2. ππ = 16, 12, 9, 6.75 3. ππ = 1, 3, 6, 10
Other Sequences:
Ex 6: Find an explicit formula for the nth term of a sequence beginning with 1
5,
4
25,
9
125,
16
625, β¦
Ex 7: Find an explicit formula for the nth term of a sequence beginning with 1, 8, 27, 64, 125, β¦ Arithmetic and Geometric Means Ex 8: Find the missing term in each arithmetic sequence
a. β¦ , 13, ____ , 7, β¦
b. β¦ , β4
7, ____, β
68
21, β¦
Ex 9: Find the missing term in each geometric sequence
a. β¦ , β3, _____ , β108, β¦
b. β¦ , β1, ____, β16, β¦
8.3 End Behavior of Sequences End Behavior Limit as n approaches infinity Diverge vs. Converge
Ex 1: Find the limit of the following sequences. Then state whether it is convergent or divergent. a. ππ = 4π + 5
b. ππ =1
π+3
c. ππ = (β1)π
d. ππ =1+4π
π
e. ππ = 6 Properties of Limits and Constants
Ex 2: Find the limit of the sequence p where ππ =5π+1
π
Properties of Limits on Operations with Sequences
Ex 3: Consider ππ =4π+3
π and ππ =
3πβ2
π
a. Find limπββ
ππ and limπββ
ππ
b. Use the properties of limits to find limπββ
(ππ β ππ)
8.4 Arithmetic Series The Sum of Terms of a Sequence Arithmetic Series Sum of an Arithmetic Series Ex 1: Find the sum of the arithmetic series 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 Ex 2: Find the sum of the arithmetic series 7 + 10 + 13 + β¦ + 55
Ex 3: Find the sum of the arithmetic series 1 + 3Β·5 + 6 + 8Β·5 + . . . + 101 Ex 4: A student borrowed $8000 for college expenses. The load was to be repaid over a 100-month period, with monthly payments as follows: $120.00, $119.50, $119.00, β¦ , $70.50
a. How much did the student pay over the life of the loan?
b. What was the total interest paid on the loan? Ex 5: The main floor of a new auditorium is planned to seat 800 people, with seats arranged in 20 rows. Each row will have 2 more seats than the previous row. How many seat should there be in the first row?
8.5/8.7 Geometric Series and Infinite Series Formula for the Sum of a Finite Geometric Series Ex 1: A geometric series is given by 4 + (4 β 3) + (4 β 32) + (4 β 33) + (4 β 34). Evaluate π5. Ex 2: A geometric series is given by 1 β 4 + 16 β 64. Evaluate π9.
Ex 3: Evaluate β 24(1
3)πβ16
π=1
Ex 4: Evaluate
Infinite Series
Ex 5: Consider the infinite sequence 4
3,
8
3,
16
3, β¦ ,
2π+1
3, β¦ and its associated series
4
3+
8
3+
16
3+ β¦ +
2π+1
3+. ..
a. What type of sequence is this? Justify your answer.
b. Find the first five partial sums of the series.
c. Does the infinite series seem to converge? Formula for the Sum of a Convergent Infinite Geometric Series Ex 6: Which of the following could be a convergent geometric series? For those that could be, give the sum.
a. 5
3,
5
9,
5
27,
5
81, β¦
b. 5
3+
5
9+
5
27+
5
81+ β―
c. 640 + 160 + 40 + 10 + β―
d. 1
2+
5
2+
9
2+
13
2