Arithmetic Sequences

Lesson 5 (3Lesson 5 (3rdrd 6 6 Weeks)Weeks)

TEKS 6.4 A/BTEKS 6.4 A/B

SequenceSequence

• A set of A set of numbersnumbers written in a written in a particular particular orderorder..

– For Example: 6, 10, 14, 18 is a For Example: 6, 10, 14, 18 is a sequence of four numbers. The sequence of four numbers. The number 6 is the number 6 is the firstfirst term in the term in the sequence, sequence, 1010 is the second term, is the second term, 1414 is the third term, and is the third term, and 1818 is the is the fourth term.fourth term.

Arithmetic Sequence

• A sequence of numbers where the difference between the successive terms is constant.–For Example: The first five terms of

an arithmetic sequence are 3, 6, 9, 12, 15… The number 3 is the first term in the sequence, 6 is the second term, 9 is the third term, 12 is the fourth term, and 15 is the fifth term.

• The The common differencecommon difference in an in an arithmetic sequence can be arithmetic sequence can be identified by finding the identified by finding the differencedifference between the between the termsterms in the in the sequencesequence..

3,

15,…9,

6,

12,

+3

+3

+3

+3

• In the sequence 3, 6, 9, 12, 15,… the In the sequence 3, 6, 9, 12, 15,… the common difference is common difference is 33..

3,

15,…9,

6,

12,

+3

+3

+3

+3

Follow these Follow these guidelinesguidelines to find a to find a rulerule or or expressionexpression that can be used to find the that can be used to find the

nnthth term in an term in an arithmeticarithmetic sequence: sequence:

1.1. Use the Use the common differencecommon difference to find a to find a patternpattern that shows the that shows the relationshiprelationship between the term’s between the term’s position numberposition number and the and the valuevalue of the of the termterm..

2.2. Multiply the Multiply the common differencecommon difference and and the the positionposition number. number.

3.3. Adjust by Adjust by addingadding or or subtractingsubtracting to to get the get the valuevalue of the of the termterm needed. needed.

4. State the 4. State the patternpattern as a as a rulerule..

5. 5. CheckCheck to see whether the to see whether the rulerule works works for the next two for the next two termsterms in the in the sequencesequence..

6. 6. RepresentRepresent the rule as an the rule as an algebraic algebraic expressionexpression..

Example 1:Example 1:

Position # Value of the Term (VOT)

1 32 6

3 94 125 15n ?

• The The common differencecommon difference of the “Value of the “Value of the Term” is of the Term” is 33..

• Multiply the position number by the Multiply the position number by the common differencecommon difference..

Position # Value of the Term (VOT)

1 3

2 6

3 9

4 12

5 15

n ?

+3

+3

+3

+3

1 x 3 =

2 x 3 =

3 x 3 =

4 x 3 =

5 x 3 =

n x 3

• The pattern is “multiply the position number by 3 to get the value of the term.”

• Written as an expression: n x 3 or 3n

Position # Value of the Term (VOT)

1 1 • 3 = 3 3

2 2 • 3 = 6 6

3 3 • 3 = 9 9

4 4 • 3 = 12 12

5 5 • 3 = 15 15

N n • 3 ?

Example:Example:

x y1 22 53 84 11n

+3

+3

+3

Notice the numbers in the x column are successive.

• In the sequence 2, 5, 8, 11 …, the common difference is 3.

• Multiply the common difference times the x-value. 1 x 3 = 3

• The first y-value is 2, not 3, therefore you must add or subtract from 3 to find the y-value (adjust). 3 -1 = 2

x y

1 2

2 5

3 8

4 11

n

+3

+3

+3

x 3 =

• Maybe each y-value in this sequence is equal to 3 times its x-value an subtract 1. x • 3 - 1 = y or 3x – 1 = y

• Check to see whether the rule works for the next two terms in the sequence.

2 x 3 -1 = 53 x 3 – 1 = 8

x y

1 2

2 5

3 8

4 11

n

x 3 – 1 =

x 3 – 1 =x 3 – 1 =

x 3 – 1 =

x 3 – 1 =

• Represent the rule as an algebraic expression.

3n - 1

x y

1 2

2 5

3 8

4 11

n

x 3 – 1 =

x 3 – 1 =x 3 – 1 =

x 3 – 1 =

x 3 – 1 =

Example:Example:

x y1 73 155 236 27

• Notice that the x-values are not successive until you get to the values of 5 and 6. You can only look for the common difference when the terms are successive.

+4

• Multiply 4 by the x values. Notice that when you do, you don’t get the y-values.

• So you must add or subtract to get the y-values.

• Check to see if the rule works. Then write it algebraic.

x y

1 7

3 15

5 23

6 27

+4

x 4 + 3 =x 4 + 3 =

x 4 + 3 =x 4 + 3 =

4x + 3 x 4 =