Lesson 2.2Finding the nth term

Writing the RULE for a Linear Sequence

Homework: lesson 2.2/1-8

Objectives

• Use inductive reasoning to find a pattern• Create a rule for finding any term/value in

the sequence• Use your rule to predict any term in the

sequence

Function Rule: The rule that gives the nth term for a sequence.n = term number (location of a value in the sequence)

20, 27, 34, 41, 48, 55, . . .Next term?

62200th term?

WHY? How do we find this 200th term?

n = 2007n+13 => 7(200)+13 = 1413

Looking at 1, 4, 7, 10, 13, 16, 19, ......., carefully helps us to make the following

observation:

As you can see, each term is found by adding 3, a common difference from the previous term

Common Difference:

Looking at 70, 62, 54, 46, 38, ... carefully helps us to make the following observation:

This time, to find each term, we subtract 8, a common difference from the previous

term

• Common difference (n) +/- ‘something’ n = 1 2 3 4 5 6 values = 7, 2, -3, -8, -13, -18, … -5 -5 -5 -5

• -5n +/- -5(1) = -5 + 12 = 7

• nth term RULE: -5n + 12

Common Difference

+/- something

Writing the Rule/ nth term

+ 12

Finding the nth Term

• Find the Common Difference• CD becomes the coefficient of n• add or subtract from that product to find the

sequence value +/- x• Write the RULE

n 1 2 3 4 5 … n … 25

value 3 9 15 21 27

+66n

-36n - 3

6n-36(25)-3

147

Use the Rule to complete the pattern

n 1 2 3 4 5n-3 -2 -1 0 1 2

n 1 2 3 4 52n+1 3 5 7 9 11

n 1 2 3 4 5-4n+5 1 -3 -7 -11 -15

What pattern do you see

consistently emerging from all

these rules?

Are these examples of linear or

nonlinear patterns?

Common difference

Term 1 2 3 4 5 6 7 … n … 20th

Value 7 2 -3 -8 -13 -18 -23

Function Rule: -5n + 12

20th term => -88

Common Difference = -5Adjust => -5n +/- ________+ 12

Use the pattern to find the rule & the missing term

n 1 2 3 4 5 .. 546n 6 12 18 24 30 324

+6 +6 +6 +6

RULE: 6n+ _?__Common difference = 6

n=1 6(1)+ _?__ = 6n=2 6(2)+ ? =12 ? = 0 RULE: 6n

n 1 2 3 4 5 .. 372x+5 7 9 11 13 15 79

+2 +2 +2 +2

RULE: 2n+ _?__Common difference = 2

n=1 2(1)+ _?__ = 7n=2 2(2)+ ? =9 ? = 5 RULE: 2n+5

n 1 2 3 4 5 .. 50-4n+1 -3 -7 -11 -15 -19 199

-4 -4 -4 -4

RULE: -4n+ _?__Common difference = -4

n=1 -4(1)+ _?__ = -3n=2 -4(2)+ ? =-7 ? = +1 RULE: -4n+1

Use a table to find the number of squares in the next shape in the

pattern.

n n 50# of squares

15

28

311

3n+2

152

• Rules that generate a sequence with a constant difference are linear functions.

n 1 2 3 4 5n-3 -2 -1 0 1 2

Ordered pairsxy

Rules for sequences can be expressed using function notation.

f (n) = −5n + 12

In this case, function f takes an input value n, multiplies it by −5, and adds 12 to produce an

output value.

n 1 2 3 4 5

f(n) -3 -1 3 11 27n 1 2 3 4 5f(n) 9 6 3 0 -3

n 1 2 3 4 5f(n) -8 -4 0 4 8

n 1 2 3 4 5f(n) -2 -1 1 4 8

IS THE PATTERN LINEAR?

NO YES; cd=-3

YES; cd=+4 NO

-5-33-28-23-18-13-8-32-5n + 7+321191613107413n – 2-2-11-9-7-5-3-113-2n + 5+429252117139514n – 3+13210-1-2-3-4n – 5

Difference87654321n

Copy and complete the tableTerm

Function Rule

Coefficient

• Find the next term in an Arithmetic and Geometric sequence

• Arithmetic Sequence• Formed by adding a fixed number to a previous

term

• Geometric Sequence• Formed by multiplying by a fixed number to a

previous term

Arithmetic sequence formula

dnaan 11

1ana n represents the term you are calculating

1st term in the sequence

d the common difference between the terms