Sequences, Induction and Probability

Aim #10-1: What are sequences?

Definition of a sequence:

An infinite sequence {an} is a function whose domain is the set of positive integers. The function values or terms, of the sequences are represented by a1, a2, a3, a4, …an…

Sequences whose domain consists only of the first n positive integers are called finite sequences.

Example 1:

Write the first four terms of the sequence whose nth term or general term, is given:

an 3n 4

an ( 1)n

3n 1

Check for Understanding:

Write the first four terms of the sequence whose nth term or general term , is given:

an 2n 5

an ( 1)n

2n 1

What is a Recursive Formula?

A recursive formula defines the nth term of a sequence as a function of the previous term.

Example 2: Using a Recursive Formula

Find the first four terms of the sequence in which a1=5 and an=3an-1+2 for n >2

Solution:

a1=5 first term

a2=3(5) + 2= 17

a3 =3(17) + 2 = ___ Continue substituting previous term in 3an-1

to find next term.

Check for understanding:

Find the first four terms of the sequence in which a1=3 and an=2an-1+ 5 for n >2

What is Factorial Notation?

Products of consecutive positive integers can be expressed in a special notation, called factorial notation.

Factorial notation:

If n is a positive integer, the notation n! (read “factorial”) is the product of all positive integers from n down through 1.

n! = n (n - 1)(n - 2)…

0!=(zero factorial), by definition, is 1.

Example 3: Finding Terms of a Sequence Involving Factorials

Write the first four terms of the sequence whose nth term is

Solution:

We need to find the first four terms of the sequence. To do so we replace each n, with 1, 2, 3, and 4.

an 2n

(n 1)!

Check for understanding:

Write the first four terms of the sequence whose nth term is

an 20

(n 1)!

Example 4: Evaluating Fractions with Factorials

Evaluate each factorial expression:

a.10!

2!8!

10 9 8!

2 1 8!

b.(n 1)!

n!(n 1)n!

n!

Check for Understanding

Evaluate each factorial expression:

a.14!

2!12!

b.n!

(n 1)!

What is Summation Notation?

The sum of the first n terms of a sequence is

represented by the summation notation

where i is the index of summation, n is the upper

limit of summation, and 1 is the lower limit of

summation.

ai a1 a2 a3... ani1

n

Example 5: Using Summation Notation

Expand and evaluate the sum:

a. (i2 1)i1

6

b. [( 2)k 5]k4

7

c. 3i1

5

Check for Understanding

Expand and evaluate the sum:

a. 2i2

i1

6

b. (2k 3)k3

5

c. 4i1

5

Example 6: Writing Sums

Express each sum using summation notation.

a.13 23 33 ....73

b.11

3

1

9

1

27 ...

1

3n 1

Summary: Answer in complete sentences.

What is a sequence? Give an example with your description.

Explain how to write your terms of sequence if the formula for the general term is given.

What is a recursive formula?

Explain how to find n! if n is a positive integer.

Explain the best way to evaluate without a calculator.

What is the meaning symbol for summation? Give an example of how it can be used.

900!

899!

Aim #10-2: What is an arithmetic sequence?

An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount.

The difference between consecutive terms is called the common difference of the sequence.

The common difference is found by subtracting any term from the term that directly follows it.

What is the common difference?

142, 146, 150, 154, 158, …

-5, -2, 1, 4, 7, …

8, 3, -2, -7, -12, …

What is the rule of the arithmetic sequence?

dnaan )1(1

a1 first termn 1termd common difference

Example 2: Using the Formula

Check for Understanding:

Example 3: Using the Arithmetic Sequence

Check for Understanding:

Thanks to drive-thrus and curbside delivery, American are eating more meals behind the wheel. In 2004, we averaged 32 a la car meals increasing by approximately 0.7 meals per year.

Write a formula for the nth term of the arithmetic sequence that models the average number of car meals n years after 2003.

How many car meals will Americans average by the year 2014?

The Sum of the First n Terms

The sum of the first n terms of an arithmetic sequence, denoted by Sn and called the nth partial sum.

Sn n2

(a1 an )

in which a1 is the first term and an is the nth term.

Example 4: Finding the Sum of n Terms

Find the sum of the first 100 terms of the arithmetic sequence: 1, 3, 5, 7, ….

Solution:

We use the formula for the general term of an arithmetic sequence to find the a100.

What is the common difference?

Then we can use the above formula to find the sum of the first 100 terms.

Sn n

2(a1 an )

Check for Understanding:

Find the sum of the first 15 terms of the arithmetic sequence: 3, 6, 9,12, ...

Example 5: Using Sn to Evaluate a Summation

Find the following sum:

Solution:

You can evaluate the first four terms and the last term.

Then you can use the Sn formula.

(5i 9)i1

25

Summary: Answer in complete sentences.

What is an arithmetic sequence? Give an example with your explanation.

What is the common difference in an arithmetic sequence?

Explain how to find the sum of the first n terms of an arithmetic sequence without having to add up all of the terms.

Find the sum of the first 80 positive even integers.

Aim #10-3: What is a geometric sequence?

A geometric sequence is a sequence in which you multiply a fixed number to obtain the next number in the sequence.

The amount by which we multiply each time is called the common ratio of the sequence.

The common ratio of a sequence can be found by dividing.

How do we find the common ratio?

Geometric sequence

5, 25, 125, 625, …

4, 8, 16, 32, 64, …

6, -12, 24, -48, 96, …

-3, 1, -1/3, 1/9, …

What is the common ratio?

Example 1: Writing the terms

Write the first 6 terms of the geometric sequence with the first term 6 and common ratio 1/3.

Check for Understanding:

Write the first 6 terms of the geometric sequence with the first term 12 and common ratio 1/2.

What is the general term for the geometric

sequence?

ana1rn 1

a1 is the first term

r is the common ratio

n 1 is the term

Example 2: Using the formula

Find the eighth term of the geometric sequence whose first term is -4 and whose common ratio is -2.

Check for Understanding:

Find the seventh term of the geometric sequence whose first term is 5 and whose common ratio is -3.

The Sum of the First n Terms of a Geometric Sequence

The sum, Sn , of the first n terms of a geometric sequence is given by :

in which a1 is the first term and r is the common ratio (r≠0)

Sn a1(1 rn )

1 r

Example 4: Finding the Sum of the First n Terms of a Geometric Sequence

Find the sum of the first 18 terms of the geometric sequence:

2, -8, 32, -128

Solution:

Replace n with 18

Identify the common ratio

Then evaluate the formula

Sn a1(1 rn )

1 r

Check for Understanding:

Find the sum of the first 9 terms of the geometric sequence:

2, -6, 18, -54, …

What is a geometric series?

An infinite sum of the form

a1 + a1r + a1r2 + a1r3 + …a1r n-

1+…with the first term a1 and the common ratio r is

called an infinite geometric series.

How can we determine which infinite geometric series have sums and which do not?

The Sum of an Infinite Geometric Series

Example 8: Finding the Sum of an Infinite Geometric Series

Check for Understanding:

Find the sum of the infinite geometric series:

3 + 2 + 4/ 3+ 8/9 + …

Summary:Answer in complete

sentences.

What is the geometric sequence? Give an example with your explanation.

What is the common ratio in a geometric sequence?

Find the sum of the first 11 terms of the geometric sequence: 4, -12, 36, -108, …

Find the sum of the first 14 terms of the geometric sequence:

1

24,

1

12,

1

6,1

3,...

Aim #10-5: What is the binomial

theorem?

Binomial is a polynomial with 2 terms.

Here we are going to provide a formula that provides a quick method of raising a binomial to a power.

The Binomial Theorem:

Example 1: Finding Binomial Coefficients

Example 2: Finding Binomial Coefficients

1112

112

47

37

)4

)3

)2

)1

C

C

C

C

From the last example you should have notice that:

rnnrn CC

Example 3:Expanding a Binomial Using Pascal’s Triangle

Finding the Term

Guided Practice:

Find the 6th term of (a + 2b) 8

More Guided Practice:

Find the 5th term of (4a + 3b )7.

Summary:Answer in complete sentences.

Explain how to evaluate . Provide an example.

What role of the binomial theorem does it play?

What is true about the sum of the exponents on a and b in any term in the expansion of (a + b) n?

rn