Chapter 6 Sequences And Series

Look at these number sequences carefully can you guess the next 2 numbers? What about guess the rule?

Can you work out the missing numbers in each of these sequences?

Now try these sequences – think carefully and guess the last number! , +2, +3 … double

This is a really famous number sequence which was discovered by an Italian mathematician a long time ago. It is called the Fibonacci sequence and can be seen in many natural things like pine cones and sunflowers!!! etc… Can you see how it is made? What will the next number be? 34!

Guess my rule! For these sequences I have done 2 maths functions! x x +1

What is a Number Sequence? A list of numbers where there is a pattern is called a number sequence The numbers in the sequence are said to be its members or its terms.

Sequences To write the terms of a sequence given the n th term Given the expression: 2n + 3, write the first 5 terms In this expression the letter n represents the term number. So, if we substitute the term number for the letter n we will find value that particular term. The first 5 terms of the sequence will be using values for n of: 1, 2, 3, 4 and 5 term 1 2 x term 2 2 x term 3 2 x term 4 2 x term 5 2 x

Sequences Now try these : Write the first 3 terms of these sequences: 1)n + 2 2)2n + 5 3)3n - 2 4)5n + 3 5) - 4n ) n , 4, 5 7, 9, 11 1, 4, 7 8, 13, 18 6, 2, - 2, 3, 6, 11,

6B - The General Term of A Number Sequence Sequences may be defined in one of the following ways: listing the first few terms and assuming the pattern represented continues indefinitely giving a description in words using a formula which represents the general term or n th term.

The first row has three bricks, the second row has four bricks, and the third row has five bricks. If u n represents the number of bricks in row n (from the top) then u 1 = 3, u 2 = 4, u 3 = 5, u 4 = 6,....

This sequence can be describe in one of four ways: Listing the terms: u 1 = 3, u 2 = 4, u 3 = 5, u 4 = 6,....

This sequence can be describe in one of four ways: Using Words: The first row has three bricks and each successive row under the row has one more brick...

This sequence can be describe in one of four ways: Using an explicit formula: u n = n + 2 u 1 = = 3 u 2 = = 4 u 3 = = 5 u 4 = = 6,....

This sequence can be describe in one of four ways: Using a graph

What you really need to know! An arithmetic sequence is a sequence in which the difference between any two consecutive terms, called the common difference, is the same. In the sequence 2, 9, 16, 23, 30,..., the common difference is 7.

What you really need to know! A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 256,.., the common ratio is 4.

Example 1: State whether the sequence -5, -1, 3, 7, 11, … is arithmetic. If it is, state the common difference and write the next three terms.

Example 2:Subtract Common difference 11 – – – – , -1, 3, 7, 11, Arithmetic! , 19, 23

Example 2: State whether the sequence 0, 2, 6, 12, 20, … is arithmetic. If it is, state the common difference and write the next three terms.

Example 2:Subtract Common difference 20 – – – – 0 2 0, 2, 6, 12, 20 … Not Arithmetic!

Example 3: State whether the sequence 2, 4, 4, 8, 8, 16, 16 … is geometric. If it is, state the common ratio and write the next three terms.

Example 3:Divide Common ratio 16 ÷ ÷ ÷ ÷ ÷ ÷ 2 2 2, 4, 4, 8, 8, 16, 16, … Not Geometric!

Example 4: State whether the sequence 27, -9, 3, -1, 1/3, … is geometric. If it is, state the common ratio and write the next three terms.

Example 4:Divide Common ratio 1/3 ÷ -1 -1/3 -1 ÷ 3 -1/3 3 ÷ -9 -1/3 -9 ÷ 27 -1/3 27, -9, 3, -1, 1/3, Geometric! -1/3 -1/9, 1/27, -1/81

Classwork Page 154 (6B) All 5. Homework Compare Arithmetic and Geometric Sequences

An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms.

Which of the following sequences are arithmetic? Identify the common difference. YES YES YES NO NO

T h e c o m m o n d i f f e r e n c e i s a l w a y s t h e d i f f e r e n c e b e t w e e n a n y t e r m a n d t h e t e r m t h a t p r o c e e d s t h a t t e r m. C o m m o n D i f f e r e n c e = 5

The general form of an ARITHMETIC sequence. First Term: Second Term: Third Term: Fourth Term: Fifth Term: n th Term:

Formula for the nth term of an ARITHMETIC sequence. I f w e k n o w a n y t h r e e o f t h e s e w e o u g h t t o b e a b l e t o f i n d t h e f o u r t h.

Given: Find: IDENTIFYSOLVE

Given: Find: What term number is (-169)? IDENTIFY SOLVE If it’s not an integer, it’s not a term in the sequence

Given: IDENTIFYSOLVE Find: What’s the real question?The Difference

Given: IDENTIFYSOLVE Find:

Homework Page ( Any 8 Problems) Take Home Test Due Tuesday.

G e o m e t r i c S e r i e s

Geometric Sequence The ratio of a term to it’s previous term is constant.The ratio of a term to it’s previous term is constant. This means you multiply by the same number to get each term.This means you multiply by the same number to get each term. This number that you multiply by is called the common ratio (r).This number that you multiply by is called the common ratio (r).

Example: Decide whether each sequence is geometric. 4,-8,16,-32,… -8 / 4 =-2 16 / -8 = / 16 =-2 Geometric (common ratio is -2) 3,9,-27,-81,243,… 9 / 3 =3 -27 / 9 = / -27 =3 243 / -81 =-3 Not geometric

Rule for a Geometric Sequence u n =u 1 r n-1 Example: Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,…. Then find u 8. First, find r.First, find r. r= 2 / 5 =.4r= 2 / 5 =.4 u n =5(.4) n-1u n =5(.4) n-1 u 8 =5(.4) 8-1 u 8 =5(.4) 7 u 8 =5( ) u 8 =

One term of a geometric sequence is u 4 = 3. The common ratio is r = 3. Write a rule for the nth term. Then graph the sequence. If u 4 =3, then when n=4, u n =3.If u 4 =3, then when n=4, u n =3. Use u n =u 1 r n-1Use u n =u 1 r n-1 3=u 1 (3) 4-1 3=u 1 (3) 3 3=u 1 (27) 1 / 9 =a 1 u n =u 1 r n-1u n =u 1 r n-1 u n =( 1 / 9 )(3) n-1 To graph, graph the points of the form (n,u n ).To graph, graph the points of the form (n,u n ). Such as, (1, 1 / 9 ), (2, 1 / 3 ), (3,1), (4,3),…Such as, (1, 1 / 9 ), (2, 1 / 3 ), (3,1), (4,3),…

Two terms of a geometric sequence are u 2 = -4 and u 6 = Write a rule for the nth term. Write 2 equations, one for each given term. u 2 = u 1 r 2-1 OR -4 = u 1 r u 6 = u 1 r 6-1 OR = u 1 r 5 Use these equations & sub in to solve for u 1 & r. -4 / r =u =( -4 / r )r = -4r = r 4 4 = r & -4 = r If r = 4, then u 1 = -1. u n =(-1)(4) n-1 If r = -4, then u 1 = 1. u n =(1)(-4) n-1 u n =(-4) n-1 Both Work!

6D1 (4 a and b) 5, 10, 20, 40 So, geometric sequence with u 1 = 5 r = 2

6D1 (9a) u 4 = 24 u 7 = 192

Homework Page 160 (6D.1 All) Take Home Test Due Tuesday

Compound Interest

Time(Years) Amount $ Interest 100 Interest Compounding Period Interest 121 Compound Interest - Future Value

COMPOUND INTEREST FORMULA Where FV is the Future Value in t years and PV is the Present Value amount started with at an annual interest rate r compounded n times per year.

Find the amount that results from the investment: $50 invested at 6% compounded monthly after a period of 3 years. EXAMPLE $59.83

Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year: FV = PV(1 + r) = 1,000(1 +.1) = $ COMPARING COMPOUNDING PERIODS

Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year: COMPARING COMPOUNDING PERIODS

Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year: Interest Earned

Page 165 6D.3, #1 a & b The investment will amount to $3993 B.) Interest = amount after 3 yrs – initial amount $ $3000 = $993

Page 165 6D.3, #2 a & b The investment will amount to € B.) Interest = € – € €

Page D.3 (3 – 10). (3 – 10). Homework

6E – Sigma Notation

Vocabulary maximum value of n starting value of n expression for general term Sigma – “take the sum of…” Read: “the summation from n = 1 to k of a n ”

Introduction to Sigma Notation = ∑ 3n 5 n = 1 5 ∑ 3n3n Is read as “the sum from n equals 1 to 5 of 3n.” index of summationlower limit of summation upper limit of summation How many terms given? 58

Formulas Arithmetic Sum: 59

Formulas Sigma Form of Arithmetic Series: 10/6/ :29 PM60

Example 1 Write in Sigma Notation,

Example 2 Write in Sigma Notation, /6/ :29 PM62

Your Turn Write in Sigma Notation, 4, 15, 26, …, /6/ :29 PM63

Example 3 Find the following sum:

Example 4 Find the following sum: 4, 15, 26, …, /6/ :29 PM65

Your Turn Find the following sum: 15, 11, 7, …, –61 10/6/ :29 PM66

Example 5 Evaluate 10/6/ :29 PM67

Example 5 Evaluate 10/6/ :29 PM68

Evaluate 10/6/ :29 PM69 Example 6

Evaluate 10/6/ :29 PM70 Your Turn

Write the expression in expanded form and then find the sum. = = -8

Write the expression in expanded form and then find the sum. = = 3236

Consider the Sequence a) Write down an expression for S n.

Consider the Sequence b) Find S n for n = 1, 2, 3, 4, and 5

Modeling Growth Is this an arithmetic series or geometric series? What is the common ration of the geometric series?

Sum of a Finite Geometric Series The sum of the first n terms of a geometric series is Notice – no last term needed!!!!

Formula for the Sum of a Finite Geometric Series n = # of terms a 1 = 1 st term r = common ratio What is n? What is a 1 ? What is r?

Example Find the sum of the 1 st 10 terms of the geometric sequence: 2,-6, 18, -54 What is n? What is a 1 ? What is r? That’s It!

Example: Consider the geometric series ½+…. Find the sum of the first 10 terms. Find n such that S n = 31 / 4.

log 2 32=n

Assignment

A r i t h m e t i c S e r i e s

When the famous mathematician C. F. Gauss was 7 years old, his teacher posed problem to the class and expected that it would keep the students busy for a long time. –Gauss, though, answered it almost immediately.

Suppose we want to find the sum of the numbers 1, 2, 3, 4,..., 100, that is, …+ 100

His idea was this: Since we are adding numbers produced according to a fixed pattern, there must also be a pattern (or formula) for finding the sum. –He started by writing the numbers from 1 to 100 and below them the same numbers in reverse order.

Writing S for the sum and adding corresponding terms gives: –It follows that 2S = 100(101) = 10,100 and so S = 5050.

We want to find the sum of the first n terms of the arithmetic sequence whose terms are u n = a 1 + (n – 1)d. –That is, we want to find:

Using Gauss’s method, we write: –There are n identical terms on the right side of this equation.

Find the sum of the first 40 terms of the arithmetic sequence 3, 7, 11, 15,... –Here, a = 3 and d = 4. –Using Formula 1 for the partial sum of an arithmetic sequence, we get: – S 40 = (40/2) [2(3) + (40 – 1)4] = 20( ) = 3240

W r i t e t h e f i r s t t h r e e t e r m s a n d t h e l a s t t w o t e r m s o f t h e f o l l o w i n g a r i t h m e t i c s e r i e s. W h a t i s t h e s u m o f t h i s s e r i e s ?

71 + (-27) Each sum is the same. 50 Terms

Find the sum of the terms of this arithmetic series.

Find the sum of the terms of this arithmetic series. What term is -5?

Homework 6F Page 169 (1 – 11) Chose 9 For #8, Just do a) & b).

Formula for the Sum of a Finite Geometric Series n = # of terms a 1 = 1 st term r = common ratio

Example: Consider the geometric series ½+…. Find the sum of the first 10 terms. Find n such that S n = 31 / 4.

log 2 32=n

Homework 6G.1 Page 171 (1 – 5) #2a (Conjugate denominator)

Infinite Geometric Series Consider the infinite geometric sequence What happens to each term in the series? They get smaller and smaller, but how small does a term actually get? Each term approaches 0

Partial Sums Look at the sequence of partial sums: What is happening to the sum? It is approaching It’s CONVERGING TO 1.

Here’s the Rule Sum of an Infinite Geometric Series If |r| < 1, the infinite geometric series a 1 + a 1 r + a 1 r 2 + … + a 1 r n + … converges to the sum If |r| > 1, then the series diverges (does not have a sum)

Converging – Has a Sum So, if -1 < r < 1, then the series will converge. Look at the series given by Since r =, we know that the sum is The graph confirms:

Diverging – Has NO Sum If r > 1, the series will diverge. Look at …. Since r = 2, we know that the series grows without bound and has no sum. The graph confirms:

Example Find the sum of the infinite geometric series 9 – … We know: a 1 = 9 and r = ?

You Try Find the sum of the infinite geometric series 24 – – 3 + … Since r = -½

Page 173 6G.2

Page 173 All Homework REVIEW 6A (NO CALCULATOR) REVIEW 6B (WITH CALCULATOR)