Sequences and Series (Section 9.4 in Textbook)

Sequences and Series(Section 9.4 in Textbook)

ARITHMETIC SEQUENCES

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, …

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

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, …

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

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.

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:

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

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.

GEOMETRIC SEQUENCES

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

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:

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)

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.

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)

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:

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

Sum of numbers 1-100?

SERIES AND SUMMATION

NOTATION

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

Series Formulas

Arithmetic Series

Sn = – (a1 + an)

Geometric Series

Sn = n2

a1•(1 – rn)

(1 – r)________

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.

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?

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!”

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:

Evaluating Using Summation Notation

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

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

)

Find the sum of the arithmetic sequence:

1, 2, 3, 4, ……, 80

Rewrite the sum using Sigma Notation

5 + 9 + 13 +17 + … + 85

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

Ex) Find the sum of the geometric sequence

3, 6, 12, ……, 12288

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.

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

Divergent Geometric Series

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

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

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=

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 + …

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

6 33

23

4 ...

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.

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?

Express the rational numbers as a fractions of integers

1) 7.14141414

2) -17.268268268