Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Do Now: Aim: What is an arithmetic sequence and...

Aim: Arithmetic Sequence Course: Alg. 2 & Trig.

Do Now:

Aim: What is an arithmetic sequence and series?

Find the next three numbers in the sequence 1, 1, 2, 3, 5, 8, . . .

Fibonacci sequence, first published in book titled, Liber abaci, in 1202 by Leonardo of Pisa. Dealt with reproductive rights of rabbits. Leonardo also introduced algebra in Europe from the mideast. Algebra was occasionally referred to as Ars Magna, “the Great Art”.

recursive – 1 or more of the first terms are given – all other terms are defined by using the previous terms pattern found in nature

sequence – a set of ordered numbers


Fibonacci Patterns

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one year?


Fibonacci Patterns

If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide

each by the number before it, we will find the following series of numbers:

1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666...,

8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538... approached the

Golden Ratio -


Sequence

4, 8, 12, 16, . . . .

4 4 4 4 4

If the pattern is extended, what are the next two terms?

How is this sequence different from the famous Fibonacci sequence?

4, 8, 12, 16, 20, 24, . . .

4, 8, 12, 16, 20, 24, . . .

1 2 3 4

4 8 12 16 4

n

n

positive integers

terms of sequence


Model Problem

Write the rule that can be used in forming a sequence 1, 4, 9, 16, . . . , then use the rule to find the next three terms of the sequence.

positive integers

terms of sequence 2

1 2 3 4

1 4 9 16

n

n

1, 4, 9, 16, 25, 36, 49


Model Problem

Write the first five terms of the sequence where rule for the nth term is represented by n + 2

positive integers

terms of sequence n + 26543 7


Definition of Arithmetic Sequence

A sequence is arithmetic if the differences between consecutive terms are the same. Sequence

a1, a2, a3, a4, . . . . . an, . . .

is arithmetic if there is a number d such that

a2 – a1 = d, a3 – a2 = d, a4 – a3 = d, etc.

The number d is the common difference on the arithmetic sequence. Each term after the first is the sum of the preceding term and a constant, c.

7, 11, 15, 19, . . . . 4 4 4 4 = d

2, -3, -8, -13, . . . . -5 -5 -5 -5 = d

4n + 3, . . .

7 – 5n, . . .

finite

infinite

1st term nth term


The nth Term of an Arithmetic Sequence

The nth term of an arithmetic sequence has the form

an = dn + c

where d is the common difference between consecutive terms of the sequence and

c = a1 – d

Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2.a1 = 2

an = dn + c an = 3n – 1

2 = 3(1) + can = dn + c c = -1

2, 5, 8, 11, 14, . . . , 3n – 1, . . .

An alternative form of the nth term is an = a1 + (n – 1)d

7, 11, 15, 19, . . . . 4 4 4 4 = d

4n + 3, . . .


Model Problem

Find the twelfth term of the arithmetic sequence 3, 8, 13, 18, . . . .

d = 5 an = dn + c

c = a1 – d

an = a1 + (n – 1)d

a12 = ?

a12 = 3 + (12 – 1)5

a12 = 3 + (11)5 = 58

Rule? an = dn + c c = a1 – d

c = 3 – 5 = -2an = 5n – 2

18 = 5(4) – 2 check:


Model Problem

The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first several terms of this sequence.

a13 = 9d + a4

65 = 9d + 20

a4 = 5(4) + c 20 = 5(4) + c

an = 5n + 0

1 2 3 4 5 6

5, 10, 15, 20, 25, 30, . . .

an = dn + c

5 = d

0 = c

an = dn + c


Regents Problem

Find the first four terms of the recursive sequence defined below.

a1 = -3 an = a(n – 1) – n

a2 = a(2 – 1) – 2

a2 = a(1) – 2

a2 = -3 – 2 = -5

a3 = a(3 – 1) – 3

a3 = a(2) – 3

a3 = -5 – 3 = -8

a4 = a(4 – 1) – 4

a4 = a(3) – 4

a4 = -5 – 4 = -12

-3, -5, -8, -12


Do Now:

Aim: What is an arithmetic sequence and series?

Regents Problem

What is the 10th term of the arithmetic sequence -1, 3, 7, 11, . . .


Regents Problem

What is the 10th term of the arithmetic sequence -1, 3, 7, 11, . . .

a10 = -1 + (10 – 1)4

a10 = -1 + (9)5 = 44

an = a1 + (n – 1)d

d = 4


Arithmetic Series

A series is the expression for the sum of the terms of a sequence.

finite sequence

6, 9, 12, 15, 18

infinite sequence

3, 7, 11, 15, . . .

finite series

6 + 9 + 12 + 15 + 18

infinite series

3 + 7 + 11 + 15 + . . .

a1 + a2 + a3 + a4 + a5a1, a2, a3, a4, a5

a1 + a2 + a3 + a4 + . . .a1, a2, a3, a4, . . .


Find the following sum

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

The Sum of an Arithmetic Sequence: Series

The sum of a finite arithmetic sequence with n terms is

An arithmetic series is the indicated sum of the terms of an arithmetic sequence.

)(2 1 naan

S

)(2 1 naan

S

Is this an arithmetic sequence? Why? d = 2

)191(2

10S

10 terms n = 10

a1 = 1 an = 19

100

When famous German mathematician Karl Gauss was a child, his teacher required the students to find the sum of the first 100 natural numbers. The teacher expected this problem to keep the class busy for some time. Gauss gave the answer almost immediately. Can you?


In an arithmetic series, if a1 is the first term, n is the number of terms, an is the nth term, and d is the common difference, then Sn the sum of the arithmetic series, is given by the formulas:

The Sum of an Arithmetic Sequence: Series

)(2 1 naan

S

1[2 ( 1) ]2

nS a n d

or


Model Problem

Find the sum of the first ten terms of an arithmetic sequence whose first term is 5 and whose 10th term is -13.

a1 = 5

)(2 1 naan

S

105 13 5 8 40

2S

a10 = -13


Model Problem

Find the sum of the first fifty terms of an arithmetic sequence 3 + 5 + 7 + 9 + . . . .

a1 = 5

)(2 1 naan

S

503 101 2600

2S

a50 = ?

an = dn + c

c = a1 – d

an = a1 + (n – 1)dd = 2, n = 50

an = a1 + (n – 1)d

an = 3 + (50 – 1)2 = 101


Model Problem – Option 2

Find the sum of the first fifty terms of an arithmetic sequence 3 + 5 + 7 + 9 + . . . .

a1 = 5 d = 2, n = 50

1[2 ( 1) ]2

nS a n d

50[2 3 (50 1)2] 2600

2S


Application

A small business sells $10,000 worth of products during its first year. The owner of the business has set a goal of increasing annual sales by $7,500 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years this business is in operation.

a1 = 10000

an = 7500n + 2500

d = 7500

)(2 1 naan

S

c = 10000 – 7500 = 2500

a10 = 7500(10)+ 2500 = 77500

500,437$)7750010000(2

10S


Application

A man wishes to pay off a debt of $1,160 by making monthly payments in which each payment after the first is $4 more than that of the previous month. According to this plan, how long will it take him to pay the debt if the first payment is $20 and no interest is charged?

Sn = 1160 d = 4 a1 = $20

1160 2 20 1 42

nn

1[2 ( 1) ]2

nS a n d

n = ?

40 4 42

nn

4 362

nn 22 18 1160n n

22 18 1160 0n n 2 9 580 0n n 29 20 0n n 29, 20n n


Arithmetic Means

The terms between any two nonconsecutive terms of an arithmetic sequence are called arithmetic means. In the sequence below, 38 and 49 are the arithmetic means between 27 and 60.

5, 16, 27, 38, 49, 60

arithmetic means

between 27 & 60

simple example: insert one arithmetic mean between 16 and 20 an = a1 + (n – 1)d

20 = 16 + (2)d

d = 2

an = a3 = 20a1 = 16(n – 1) = 3 – 1 = 2

16, 18, 20


Model Problem

Write an arithmetic sequence that has five arithmetic means between 4.9 and 2.5.

an = a1 + (n – 1)d

2.5 = 4.9 + (6)d

d = -0.4

an = a7 = 2.5

a1 = 4.9

n = 7

4.9, ___, ___, ___, ___, ___, 2.5

a2 = 4.9 + (-0.4) = 4.5

4.5

a3 = 4.5 + (-0.4) = 4.1a4 = 4.1 + (-0.4) = 3.7

a5 = 3.7 + (-0.4) = 3.3

a6 = 3.3 + (-0.4) = 2.9

4.1 3.7 3.3 2.9


Model Problem

Mrs. Gonzales sells houses and makes a commission of $3750 for the first house sold. She will receive a $500 increase in commission for each additional house sold. How many houses must she sell to reach total commissions of $65000?

Sn = 65000 a1 = 3750 d = 500

)(2 1 naan

S

an = a1 + (n – 1)d)3750(

265000 na

n

)500)1(3750(3750(2

65000 nn

25007000130000 nn

n = 10.58 and –24.58

She must sell 11 houses.

Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Do Now: Aim: What is an arithmetic sequence and...

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Transcript of Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Do Now: Aim: What is an arithmetic sequence and...