Chapter 14 14.1 Sequences and Series

A Sequence is an ordered set of numbers Definition Sequence A Sequence is an ordered set of numbers Definition Infinite Sequence An Infinite Sequence is an ordered set of numbers that continues on forever Ex: 3, 6, 7, 9, … Each number in a sequence is called a term

Definition Nth term The Nth term of a sequence is a rule that describes all the terms of a sequence and is sometimes referred to as the General Term of the sequence Definition Recursive Formula The Recursive definition of a sequence is a rule that describes the next term in the sequence based on the previous terms

The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.

n an DOMAIN: 1 2 3 4 5 The domain gives the relative position of each term. The range gives the terms of the sequence. RANGE: 3 6 9 12 15 This is a finite sequence having the rule an = 3n, where an represents the nth term of the sequence.

Write the first six terms of the sequence an = 2n + 3. Writing Terms of Sequences Write the first six terms of the sequence an = 2n + 3. SOLUTION a 1 = 2(1) + 3 = 5 1st term a 2 = 2(2) + 3 = 7 2nd term a 3 = 2(3) + 3 = 9 3rd term a 4 = 2(4) + 3 = 11 4th term a 5 = 2(5) + 3 = 13 5th term a 6 = 2(6) + 3 = 15 6th term

f (1) = (–2) 1 – 1 = 1 f (2) = (–2) 2 – 1 = –2 f (3) = (–2) 3 – 1 = 4 Writing Terms of Sequences Write the first six terms of the sequence f (n) = (–2) n – 1 . SOLUTION f (1) = (–2) 1 – 1 = 1 1st term f (2) = (–2) 2 – 1 = –2 2nd term f (3) = (–2) 3 – 1 = 4 3rd term f (4) = (–2) 4 – 1 = – 8 4th term f (5) = (–2) 5 – 1 = 16 5th term f (6) = (–2) 6 – 1 = – 32 6th term

If the terms of a sequence have a recognizable pattern, Writing Rules for Sequences If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the n th term of the sequence. Describe the pattern, write the next term, and write a rule for the n th term of the sequence __ – , , – , , …. 1 3 9 27 81 _

Writing Rules for Sequences SOLUTION n 1 2 3 4 1 243 - 5 1 3  , 9 27 81 terms rewrite terms 1 3 - 4 , 2 1 3 - 5 1 3 - A rule for the nth term is an = n

n Describe the pattern, write the next term, and write Writing Rules for Sequences Describe the pattern, write the next term, and write a rule for the n th term of the sequence. SOLUTION 2, 6, 12 , 20,…. n 1 2 3 4 5 30 terms 2 6 12 20 rewrite terms 1(1 +1) 2(2 +1) 3(3 +1) 4(4 +1) 5(5 +1) A rule for the nth term is f (n) = n (n+1).

You can graph a sequence by letting the horizontal axis Graphing a Sequence You can graph a sequence by letting the horizontal axis represent the position numbers (the domain) and the vertical axis represent the terms (the range).

• Write a rule for the number of oranges in each layer. Graphing a Sequence You work in the produce department of a grocery store and are stacking oranges in the shape of square pyramid with ten layers. • Write a rule for the number of oranges in each layer. • Graph the sequence.

From the diagram, you can see that an = n 2 Graphing a Sequence SOLUTION The diagram below shows the first three layers of the stack. Let an represent the number of oranges in layer n. n 1 2 3 an 1 = 1 2 4 = 2 2 9 = 3 2 From the diagram, you can see that an = n 2

an = n2 Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100). Graphing a Sequence an = n2 Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).

When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite. FINITE SEQUENCE FINITE SERIES 3, 6, 9, 12, 15 3 + 6 + 9 + 12 + 15 INFINITE SEQUENCE INFINITE SERIES 3, 6, 9, 12, 15, . . . 3 + 6 + 9 + 12 + 15 + . . . . . . You can use summation notation to write a series. For example, for the finite series shown above, you can write 3 + 6 + 9 + 12 + 15 =  3i 5 i = 1

 3i 5 i = 1 3 + 6 + 9 + 12 + 15 =  3i upper limit of summation Is read as “the sum from i equals 1 to 5 of 3i.” 5 i = 1  3i 3 + 6 + 9 + 12 + 15 =  3i 5 i = 1 index of summation lower limit of summation

USING SERIES Summation notation is also called sigma notation because it uses the uppercase Greek letter sigma, written . Summation notation for an infinite series is similar to that for a finite series. For example, for the infinite series shown earlier, you can write: 3 + 6 + 9 + 12 + 15 + =  3i  i = 1 . . . The infinity symbol,  , indicates that the series continues without end.

The index of summation does not have to be i. Any letter can be used. Also, the index does not have to begin at 1.

Writing Series with Summation Notation Write the series with summation notation. 5 + 10 + 15 + + 100 . . . SOLUTION Notice that the first term is 5 (1), the second is 5 (2), the third is 5 (3), and the last is 5 (20). So the terms of the series can be written as: ai = 5i where i = 1, 2, 3, . . . , 20 The summation notation is  5i. 20 i = 1

Writing Series with Summation Notation Write the series with summation notation. . . . 1 2 3 4 2 3 4 5 + + + + SOLUTION Notice that for each term the denominator of the fraction is 1 more than the numerator. So, the terms of the series can be written as: ai = where i = 1, 2, 3, 4 . . . i i + 1  The summation notation for the series is  i = 1 i i + 1 .

Writing Series with Summation Notation The sum of the terms of a finite sequence can be found by simply adding the terms. For sequences with many terms, however, adding the terms can be tedious. Formulas for finding the sum of the terms of three special types of sequences are shown next.

 1 = n  i =  i 2 = n (n + 1) n (n + 1)(2 n + 1) n 2 6 Writing Series with Summation Notation CONCEPT SUMMARY FORMULAS FOR SPECIAL SERIES n i = 1  1 = n gives the sum of n 1’s . 1  i = n (n + 1) 2 n i = 1 gives the sum of positive integers from 1 to n . 2  i 2 = n (n + 1)(2 n + 1) 6 n i = 1 gives the sum of squares of positive integers from 1 to n. 3

Using a Formula for a Sum RETAIL DISPLAYS How many oranges are in a square pyramid 10 layers high?

Using a Formula for a Sum SOLUTION You know from the earlier example that the i th term of the series is given by ai = i 2, where i = 1, 2, 3, . . . , 10. 10  i = 1 i 2 = 12+ 22 + + 102 . . . = 6 10(10 + 1)(2 • 10 + 1) 10(11)(21) = 6 = 385 There are 385 oranges in the stack.

HW #14.1a Pg 615-616 1-20

Find the nth term of the given sequence

Chapter 14 14.1 Sequences and Series

Write down the first five terms of the following recursively defined sequence. Find an expression for the nth term of the sequence a1 = 2 a2 = 3 + 2 = 5 a3 = 3 + 5 = 8 a4 = 3 + 8 = 11 a5 = 3 + 11 = 14

Write down the first five terms of the following recursively defined sequence. Find an expression for the nth term of the sequence

Definition Partial Sum The Partial Sum (Sn) of an infinite series is the sum of the first n terms where n is some natural number For the series given find the first 6 partial sums and determine an expression to find the nth partial sum

In the diagram, suppose each small square has sides 1 unit long, and suppose that the total height and the total width of the entire figure are each n units. Use the area formula for a triangle to find the area of the entire figure (as the sum of the areas of 1 large triangle and n small triangles), in terms of n.

In the diagram, suppose each small square has sides 1 unit long, and suppose that the total height and the total width of the entire figure are each n units. Explain how your answer to part A proves the formula for 1 + 2 + 3 + 4 + … + n

HW #14.1b Pg 616 21-44

14-2 Arithmetic Sequences Chapter 14 14-2 Arithmetic Sequences

Definition Arithmetic Sequence A sequence in which a constant d can be added to each term to get the next term is called an arithmetic sequence. The constant d is called the common difference.

To decide whether a sequence is arithmetic, find the differences of consecutive terms. an – an-1

To decide whether a sequence is arithmetic, find the differences of consecutive terms. an – an-1 3 5 7 9 4 4 4 4

Decide whether the sequence is arithmetic.

If a sequence is arithmetic we can Simplify the series Create formulas for the nth term Find the sum of the first n terms

Substituting to get to an Common Difference Recursive Substituting to get to an 3 10 7 17 24 …

a1 = a1 a2 = a1 + d a3 = (a1 + d) + d = a1 + 2d The first term of an arithmetic sequence is a1. We add d to get the next term, a1 + d. We add d again to get the next term, (a1 + d) + d, and so on. There is a pattern. a1 = a1 a2 = a1 + d a3 = (a1 + d) + d = a1 + 2d a4 = (a1 + 2d) + d = a1 + 3d … an = a1 + (n - l)d

The sequence is arithmetic, a1 = 50 and common difference d = -6.

The sequence is arithmetic, a1 = 2, d = 9

Write a rule for the nth term of the sequence and find a15

Find for the sequence For the sequence what term is 2? An arithmetic sequence has and , what is ?

Proof Theorem 14-2 Note: and All the pairs have a sum Therefore:

Proof Theorem 14-3 Continuing from 14-2:

Alternate Proof Theorem 14-3 This is the sum of the first n-1 integers. The sum of the first n integers =

HW #14.2 Pg 622-623 1-35 Odd, 36-41

Chapter 14 14-3 Geometric Sequences

Definition Geometric Sequence A sequence in which a constant r can be multiplied by each term to get the next term is called a geometric sequence. The constant r is called the common ratio.

To decide whether a sequence is geometric, find the ratios of consecutive terms. Decide whether each sequence is geometric.

Decide whether the sequence is arithmetic, geometric, or neither Decide whether the sequence is arithmetic, geometric, or neither. Explain your answer.

Finding the Nth Term Consider the sequence Define it Recursively 3, 6, 12, 24, … Define it Recursively a1 = 3 a2 = 3(2) a3 = 3(2)(2) a4 = 3(2)(2)(2) an = 2an-1 Define it Explicitly an = 3(2)n-1

Write a rule for the nth term of the sequence -8, -12, -18, -27, . . . . Then find a8.

Write a rule for the nth term of the sequence 6, 24, 96, 384, . . . . Then find a7. a1 = 6

Write a rule for the nth term of the geometric sequence. Then find a8.

A ping pong ball is dropped from a height of 16 feet and it always rebounds ¼ of the distance of the previous fall. a. What distance does it rebound after the sixth time? b. What is the total distance the ball travels after the this time?

HW #14.3 Pg 628-629 1-31 Odd, 32-40

14-4 Infinite Geometric Sequences Chapter 14 14-4 Infinite Geometric Sequences

What is happening to the sum as n gets larger?

Definition Geometric Sequence If, in an infinite series, Sn approaches some limit as n becomes very large, that limit is defined to be the sum of the infinite series. If an infinite series has a sum, it is said to converge or to be convergent

This infinite series will have a finite sum since |r| < 1 Consider the following two sums, which one will have a finite sum and why? This infinite series will have a finite sum since |r| < 1

Theorem 14-6 An infinite geometric series is convergent and thus has a sum if and only if |r| < 1 Theorem 14-7 The sum of an infinite geometric series with |r| < 1 is given by

HW #14.4 Pg 632-633 1-21 Odd, 23-27

14-5 Mathematical Induction Chapter 14 14-5 Mathematical Induction

Value 1 + 3 4 1 + 3 +5 9 1 + 3 +5 + 7 16 … 1 + 3 +5 + 7 + … (2n – 1) Consider the first n odd numbers: 1, 3, 5, 7, 9, …, 2n - 1 Partial Sum Value S1 = 1 S2 = 1 + 3 4 S3 = 1 + 3 +5 9 S4 = 1 + 3 +5 + 7 16 … Sn =. 1 + 3 +5 + 7 + … (2n – 1) n2 We have reasoned inductively that the sum of the first n odd numbers is n2, but we have not proven that to be true.

The Principle of Mathematical Induction In Lesson 14-2 we learned that the rule for the sum of the first n positive integers is: Statements like these can be proven using a method of proof called mathematical induction. The Principle of Mathematical Induction If, Pn is a statement concerning positive integers n and P1 is true, and b) assuming Pk is true implies that Pk+1 is true, then Pn must be true for all positive integers n.

HW #14.5a Pg 636 1-3

14-5 Mathematical Induction Chapter 14 14-5 Mathematical Induction

HW #14.5b Pg 636 5, 6, 10-16

HW #R-14a Pg 639-640 1-23

Test Review

After college George got a job working as an aerospace engineer After college George got a job working as an aerospace engineer. His starting salary was $40,000 a year and he got a 3% raise every year after that. How much money did he make the 32nd year he worked there. If he retires after working for 40 years, how much money would he have made total?

b. In the diagram, there is one L-shaped region corresponding to each positive integer through n. To calculate the area of the nth such region, calculate the areas of the rectangles A, B, and C shown, in terms of n, and simplify your answer. (Hint: Note that one side of each of the regions A and B has length 1 + 2 + 3 + … + n –1.)

HW #R-14b Pg 641 1-20