1. Lesson 2.2 Finding the nth term
Writing the RULE for a Linear Sequence
Homework: lesson 2.2/1-8

2. 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

3. Next term? 62
200th term?
20, 27, 34, 41, 48, 55, . . .
How do we find this 200th term?
WHY?
Function Rule:
The rule that gives the nth term for a sequence.
n = term number (location of a value in the sequence)

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

8. Looking at 1, 4, 7, 10, 13, 16, 19, ......., carefully helps us to make the following observation:
Common Difference:
As you can see, each term is found by adding 3, a common difference from the previous term

9. 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

10. Writing the Rule/ nth term
Common difference (n) +/- ‘something’
n =
values = 7, 2, -3, -8, -13, -18, …
-5n +/-
-5(1) = = 7
nth term RULE: -5n + 12
+ 12
Common Difference
+/-
something

11. Finding the nth Term n +6 Find the Common Difference1 2 3 4 5 … 25
value 9 15 21 27
6(25)-3
6n-3
147 +6
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 6n -3
6n - 3

12. Use the Rule to complete the pattern
What pattern do you see consistently emerging from all these rules? n 1 2 3 4 5
n-3 -2 -1 n 1 2 3 4 5
2n+1 7 9 11
Common difference n 1 2 3 4 5
-4n+5 -3 -7
-11
-15
Are these examples of linear or nonlinear patterns?

13. Adjust => -5n +/- ________
Term 1 2 3 4 5 6 7 … n
20th
Value -3 -8
-13
-18
-23
Common Difference = -5
Adjust => -5n +/- ________
+ 12
Function Rule:
-5n + 12
20th term => -88

14. Use the pattern to find the rule & the missing term1 2 3 4 5 .. 54 6n 6 12 18 24 30
324 +6 +6 +6 +6
RULE: 6n+ _?__
Common difference = 6
n=1  6(1)+ _?__ = 6
n=2  6(2)+ ? =12
? = 0
RULE: 6n

15. n 1 2 3 4 5 .. 37
2x+5 7 9 11 13 15 79 +2 +2 +2 +2
RULE: 2n+ _?__
Common difference = 2
n=1  2(1)+ _?__ = 7
n=2  2(2)+ ? =9
? = 5
RULE: 2n+5

16. n 1 2 3 4 5 .. 50 -3 -7 -11 -15 -19 199 -4n+1 RULE: -4n+ _?__
Common difference = -4
RULE: -4n+ _?__
n=1  -4(1)+ _?__ = -3
n=2  -4(2)+ ? =-7
? = +1
RULE: -4n+1

17. Use a table to find the number of squares in the next shape in the pattern.50
# of squares 1 2 3 5 8 11
3n+2
152

18. Rules that generate a sequence with a constant difference are linear functions.
Ordered pairs x y n 1 2 3 4 5
n-3 -2 -1

19. Rules for sequences can be expressed using function notation
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.

20. IS THE PATTERN LINEAR? n 1 2 3 4 5 f(n) -3 -1 11 27 n 1 2 3 4 5 f(n) 96 -3 NO
YES; cd=-3 n 1 2 3 4 5
f(n) -8 -4 8 n 1 2 3 4 5
f(n) -2 -1 8
YES; cd=+4 NO

22. Copy and complete the table
Term n 1 2 3 4 5 6 7 8
Difference
n – 5 -4 -3 -2 -1 1 2 3 +1
4n – 3 1 5 9 13 17 21 25 29 +4
-2n + 5 3 1 -1 -3 -5 -7 -9
-11 -2
3n – 2 1 4 7 10 13 16 19 21 +3
-5n + 7 2 -3 -8
-13
-18
-23
-28
-33 -5
Function Rule
Coefficient

24. 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

26. Arithmetic sequence formula
n represents the term you are calculating
1st term in the sequence
d the common difference between the terms