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Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d)...

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Sec 2.1.5 How Arithmetic Sequences Work? Generalizing Arithmetic Sequences
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Page 1: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Sec 2.1.5

How Arithmetic Sequences

Work?

Generalizing Arithmetic Sequences

Page 2: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Blast from the past

• Solve the system of equations:

x+9y=33

x+21y=-3

Page 3: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Test 2

Thursday Oct 31

Happy Halloween!

Page 4: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

So far:

• Linear function:

• Constant increase or

decrease.

• Same value is added

(or subtracted) to the

output as the input

increases by one unit.

• Exponential function:

• Constant growth or

decay by a common

ratio.

• The output is

multiplied (or divided)

by a common ratio as

the input increases by

one unit.

Page 5: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Next few lessons

Arithmetic sequences:

Constant increase or

decrease.

Geometric Sequences:

Constant growth or decay

by a common ratio.

Page 6: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use
Page 7: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Some terms you should know

before we start

• Definition of Counting Numbers

• The numbers which are used for counting from

one to infinity are called Counting Numbers.

• More about Counting Numbers

• Counting numbers are also called as natural

numbers.

• Counting numbers are designated as n.

Page 8: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Example on Counting Numbers

Identify the counting numbes.

A. 30

B. 9.1

C. 0

D. 10

E. -2

F. 1

Page 9: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

A sequence can be thought of as a function, with

the input numbers consisting of the natural

numbers, and the output numbers being the

terms.

Page 10: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

A sequence in which a constant (d) can be added to each term to get the next term is called an

Arithmetic Sequence.

The constant (d) is called the

Common Difference.

Page 11: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

To find the common difference (d), subtract any term from one that follows it.

2 5 8 11 14

3 3 3 3

t1 t2 t3 t4 t5

Page 12: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Find the first term and the

common difference of each

arithmetic sequence.

1.) 4,9,14,19,24First term (a): 4 Common difference (d): 2 1a a = 9 – 4 = 5

2.) 34,27,20,13,6, 1, 8,.... First term (a): 34 Common difference (d): -7

BE CAREFUL: ALWAYS CHECK TO MAKE

SURE THE DIFFERENCE IS THE SAME

BETWEEN EACH TERM !

Page 13: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Now you try!

Find the first term and the common difference of each of

these arithmetic sequences.

b) 11, 23, 35, 47, ….

a) 1, -4, -9, -14, ….

d) s-4, 3s-3, 5s-2, 7s-1, …..

c) 3x-8, x-8, -x-8, -3x-8

Page 14: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

b) 11, 23, 35, 47, ….

a) 1, -4, -9, -14, ….

d) s-4, 3s-3, 5s-2, 7s-1, …..

c) 3x-8, x-8, -x-8, -3x-8

Answers with solutions

a = 1 and

d = a2 - a1 = - 4 - 1 = - 5

a = 11 and

d = a2 - a1 = 23 - 11 = 12

a = 3x-8 and

d = a2 - a1 = x – 8 – (3x – 8) = - 2x

a = s - 4 and

d = a2 - a1 = 3s – 3 – (s – 4) = 2s + 1

Page 15: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use to give use any term that we need without listing the whole sequence .

3, 7, 11, 15, …. We know a = 3 and d = 4

t1= a = 3

t2= a+d = 3+4 = 7

t3= a+d+d = a+2d = 3+2(4) = 11

t4 = a+d+d+d = a+3d = 3+3(4) = 15

Page 16: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use to give use any term that we need without listing the whole sequence .

The nth term of an arithmetic sequence is given by:

The last # in the

sequence/or the #

you are looking for

First

term

The position

the term is in The common

difference

tn = t1 + (n – 1) d

Page 17: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Explicit Formula of a Sequence

• A formula that allows direct computation of

any term for a sequence a1, a2, a3, . . . , an, .

. . .

• To determine the explicit formula, the

pervious term need not be computed.

Page 18: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Find the 14th term of the arithmetic sequence

4, 7, 10, 13,……

(14 1) 4

4 (13)3

4 39

43

tn = t1 + (n – 1) d

t14 = 3 You are

looking for

the term!

The 14th term in this sequence

is the number 43!

Page 19: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Now you try! Find the 10th and 25th term given the following information. Make sure to derive

the general formula first and then list ehat you have been provided.

b) x+10, x+7, x+4, x+1, ….

a) 1, 7, 13, 19 ….

d) The second term is 8 and the common difference is 3

c) The first term is 3 and the common difference is -21

Page 20: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

b) x+10, x+7, x+4, x+1,.

a) 1, 7, 13, 19 …. ….

d) The second term is 8

and the common

difference is 3

c) The first term is 3 and the

common difference is -21

Answers with solutions a = 1 and d = a2 - a1 = 7 – 1 = 6

tn=a+(n-1)d = 1 + (n-1) 6 = 1+6n-6 So tn = 6n-5

t10 = 6(10) – 5 = 55

t25 = 6(25)-5 = 145

a = x+10 and d = a2 - a1 = x+7-(x+10) = -3

tn=a+(n-1)d = x+10 + (n-1)(-3) = x+10-3n+3 So tn= x-3n+13

t10 = x -3(10)+13 = x - 17

t25 = x -3(25)+13 = x - 62

a = 3 and d = -21

tn=a+(n-1)d = 3 + (n-1) -21 = 3-21n+21 So tn= 24-21n

t10 = 24-21(10) = -186 t25 = 24-21(25) = -501

a = 8 - 3 = 5 and d = 3

tn=a+(n-1)d = 5 + (n-1) 3 = 5+3n-3 So tn = 3n+2

t10 = 3(10) +2 = 32 t25 = 3(25)+2 = 77

Page 21: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Find the 14th term of the arithmetic sequence with first term of 5 and the common difference is –6.

(14 1)

tn = a + (n – 1) d t14 =

You are looking for the

term! List which variables

from the general term are

provided!

The 14th term in this sequence

is the number -73!

a = 5 and d = -6

5 -6

= 5 + (13) * -6

= 5 + -78 = -73

Page 22: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?

301 4 ( 1)3n 301 4 3 3n

301 1 3n 300 3n100 n

tn = t1 + (n – 1) d

You are

looking

for n!

The 100th term in this

sequence is 301!

Page 23: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

In the arithmetic sequence 4,7,10,13,…,

Can a term be 560?

tn = t1 + (n – 1) d

You are

looking

for n!

560 is not a term.

Page 24: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

In an arithmetic sequence, term 10 is 33 and term 22 is –3. What are the first four terms of the sequence?

The sequence is 60, 57, 54, 51, …….

Use what you know! t10=33

t22= -3

tn = t1 + (n – 1) d

For term 10: 33= a + 9d

tn = t1 + (n – 1) d

For term 22: -3= a + 21d

HMMM! Two equations you can solve!

33 = a+9d

-3 = a+21d

By elimination -36 = 12d

-3 = d

SOLVE: SOLVE: 33 = a + 9d

33 = a +9(-3)

33 = a –27

60 = a

Page 25: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

Recursive Formula

• For a sequences a1, a2, a3, . . . , an, . . . a

recursive formula is a formula that requires

the computation of all previous terms in

order to find the value of an .

Page 26: Sec 2.1.5 How Arithmetic Sequences Work?The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use

homework

• Review and Preview

• Page 78

• #71-77 all

• For additional

resources use this

page:

• http://mathbits.com/M

athBits/TISection/Alg

ebra2/sequences.htm


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