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Sequences and Series (Section 9.4 in Textbook)

Sequences and Series (Section 9.4 in Textbook)

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Page 1: Sequences and Series (Section 9.4 in Textbook)

Sequences and Series(Section 9.4 in Textbook)

Page 2: Sequences and Series (Section 9.4 in Textbook)

ARITHMETIC SEQUENCES

Page 3: Sequences and Series (Section 9.4 in Textbook)

How are these sets related?

• 3, 6, 9, 12, 15, 18, 21, 24, …

• 25, 20, 15, 10, 5, 0, -5, -10, …

• -21, 3, 27, 51, 75, 99, 123, …

Page 4: Sequences and Series (Section 9.4 in Textbook)

Arithmetic Sequence

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

Recursive Formula:

26, 21, 16, 11, 6, . . . Common Difference = 5

an = an-1 + d

Page 5: Sequences and Series (Section 9.4 in Textbook)

Is the given sequence arithmetic? If so, identify the common difference.

1. 2, 4, 8, 16, …2. 4, 6, 12, 18, 24, …3. 2, 5, 7, 12, … 4. 48, 45, 42, 39, …5. 1, 4, 9, 16, …6. 10, 20, 30, 40, …

Page 6: Sequences and Series (Section 9.4 in Textbook)

Arithmetic Sequence Explicit Formula

The “nth” number in the sequence.

Ex. a5 is the 5th number in the

sequence.

The 1st number in the sequence.

The same as the n in an. If you’re

looking for the 5th number in the

sequence, n = 5.

The common difference.

an = a1 + (n – 1) • d

Page 7: Sequences and Series (Section 9.4 in Textbook)

Examples:

1) Given the sequence -4, 5, 14, 23, 32, 41, 50,…, find the 14th term.

an = a1 + (n – 1) • d

2) Given the sequence 79, 75, 71, 67, 63,…, find the term number that is -169.

Page 8: Sequences and Series (Section 9.4 in Textbook)

For the arithmetic sequence: -5, -2, 1, 4,… find:

• The common difference:

• The 10th term:

• Recursive rule for the nth term:

• Explicit rule for the nth term:

Page 9: Sequences and Series (Section 9.4 in Textbook)

Example:

Suppose you are saving up for a new gaming system. You have 100 dollars this year, and you plan to add 33 dollars each of the following years. How much money will you have in 7 years?

an = a1 + (n – 1) • d

Page 10: Sequences and Series (Section 9.4 in Textbook)

Constructing Sequences

The 4th and 7th terms of an arithmetic sequence are -8 and 4, respectively. Find the 1st term and a explicit rule for the nth term.

Page 11: Sequences and Series (Section 9.4 in Textbook)

GEOMETRIC SEQUENCES

Page 12: Sequences and Series (Section 9.4 in Textbook)

Geometric sequences are different. See if you can spot the relationship!

• 3, 6, 12, 24, 48, 96,…

• 81, 27, 9, 3, 1, ⅓,…

• -2, 4, -8, 16, -32, 64, -128

Page 13: Sequences and Series (Section 9.4 in Textbook)

Geometric Sequences

An geometric sequence is defined as a sequence in which there is a common ratio between consecutive terms.

Common Ratio = 2

,...320,160,80,40,20,10,5

an = an-1 r Recursive Formula:

Page 14: Sequences and Series (Section 9.4 in Textbook)

The 1st number in the sequence.

Geometric Sequence Formula

The “nth” number in the sequence.

Ex. a5 is the 5th number in the

sequence.

The common ratio.

The same as the n in an. If you’re looking for the 5th number in the

sequence, n = 5.

an = a1 • r (n-1)

Page 15: Sequences and Series (Section 9.4 in Textbook)

Examples:

1) Given the sequence 4, 28, 196, 1372, 9604,…, find the 14th term.

an = a1 • r (n-1)

2) Given the sequence 1, 5, 25, 125, 625,…, find the term number that is 9,765,625.

Page 16: Sequences and Series (Section 9.4 in Textbook)

Example :Suppose you want a reduced copy of a photograph.

The actual length of the photograph is 10 in. The smallest size the copier can make is 64% of the original. Find the length of the photograph after five reductions.

an = a1 • r (n-1)

Page 17: Sequences and Series (Section 9.4 in Textbook)

The common ratio:

The 8th term:

Recursive rule for the nth term:

Explicit rule for the nth term:

You Try! For the geometric sequence: 1, -2, 4, -8, 16,… find:

Page 18: Sequences and Series (Section 9.4 in Textbook)

Fibonacci Sequence

The Fibonacci sequence can be defined recursively by:a1 = 1; a2 = 1; an = an-2 + an-1 (For all positive integers n ≥ 3)

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …Link:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

Page 19: Sequences and Series (Section 9.4 in Textbook)

Sum of numbers 1-100?

Page 20: Sequences and Series (Section 9.4 in Textbook)

SERIES AND SUMMATION

NOTATION

Page 21: Sequences and Series (Section 9.4 in Textbook)

Series

A series is the sum of the terms in a sequence.

3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135

3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135

Page 22: Sequences and Series (Section 9.4 in Textbook)

Series Formulas

Arithmetic Series

Sn = – (a1 + an)

Geometric Series

Sn = n2

a1•(1 – rn)

(1 – r)________

Page 23: Sequences and Series (Section 9.4 in Textbook)

1) Evaluate a series with the terms 1, 7, 13, 19, 25 for the first 13 terms.

2) Find the sum of the first 10 terms of the geometric series with a1 = 6 and r = 2.

Page 24: Sequences and Series (Section 9.4 in Textbook)

A philanthropist donates $50 to the SPCA. Each year, he pledges to donate 12 dollars more than the previous year. In 8 years, what is the total amount he will have donated?

Page 25: Sequences and Series (Section 9.4 in Textbook)

Summation Notation

Instead of saying: “Find the sum of the series denoted by an = 3n + 2 from the 3rd term to 7th term,” mathematicians made up a symbol to deal with it. Sigma!

∑ I’m just a fancy way of saying, “Add everything up!”

Page 26: Sequences and Series (Section 9.4 in Textbook)

sequence formula

last term

first term

7

3

23n

n

“Find the sum of the series denoted by an = 3n + 2 from the 3rd term to 7th term”

now looks like:

Page 27: Sequences and Series (Section 9.4 in Textbook)

Evaluating Using Summation Notation

Page 28: Sequences and Series (Section 9.4 in Textbook)

Ex) Use summation notation to write each series for the specified number of terms.3 + 8 + 13 + 18 + …; n = 9

Page 29: Sequences and Series (Section 9.4 in Textbook)

Sum of a Finite Arithmetic Sequence

Let {a1, a2, a3, ….} be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is

ak a1 a2 .... ank1

n

n(a1 an2

)

Page 30: Sequences and Series (Section 9.4 in Textbook)

Find the sum of the arithmetic sequence:

1, 2, 3, 4, ……, 80

Page 31: Sequences and Series (Section 9.4 in Textbook)

Rewrite the sum using Sigma Notation

5 + 9 + 13 +17 + … + 85

Page 32: Sequences and Series (Section 9.4 in Textbook)

Sum of a Finite Geometric Sequence

Let {a1, a2, a3, ….., an} be a finite geometric sequence with common ratio r ≠ 1.

Then the sum of the terms of the sequence is

ak a1 a2 ... ank1

n

a1(1 r

n )

1 r

Page 33: Sequences and Series (Section 9.4 in Textbook)

Ex) Find the sum of the geometric sequence

3, 6, 12, ……, 12288

Page 34: Sequences and Series (Section 9.4 in Textbook)

Infinite Geometric Series

Geometric series are special. Sometimes we can find their sum, even if they go on forever. In order to do this, we need to decide if the series is convergent or divergent.

Page 35: Sequences and Series (Section 9.4 in Textbook)

Convergent Geometric Series

The following series are convergent because the terms eventually approach 0.

,...8

1,4

1,2

1,1,2,4,8)1

,...001.,01.,1.,1,10,100)3

Page 36: Sequences and Series (Section 9.4 in Textbook)

Divergent Geometric Series

The following series are divergent because the terms do not have a limit.

...135,45,15,5)2

Page 37: Sequences and Series (Section 9.4 in Textbook)

Sum of an Infinite Geometric Series

The geometric series

Converges if and only if . If it does converge, the sum is

r 1

a

(1 r)

a r k 1k1

S=

Page 38: Sequences and Series (Section 9.4 in Textbook)

Examples

Determine whether each infinite geometric series diverges or converges. If it converges, find the sum.a) 1 – 1/3 + 1/9 - …

b) 4 + 8 + 16 + …

Page 39: Sequences and Series (Section 9.4 in Textbook)

Determine whether the infinite geometric series converges. If it does, find the sum.

6 33

23

4 ...

Page 40: Sequences and Series (Section 9.4 in Textbook)

Rational NumbersRepeating decimals are considered rational

numbers because they can be represented as a ratio of two integers.

A number is rational if you can write it in a form a/b where a and b are integers, b not zero.

Page 41: Sequences and Series (Section 9.4 in Textbook)

Since 0.11111... = 1/9, then the decimal number 0.11111... is a rational number.

In fact, every non-terminating decimal number that REPEATS a certain pattern of digits, is a rational number.

For example, let's make up a decimal number 0.135135135135135... that never ends. Do you believe we CAN write it as a fraction, in the form a/b?

Page 42: Sequences and Series (Section 9.4 in Textbook)

Express the rational numbers as a fractions of integers

1) 7.14141414

2) -17.268268268