Lesson 2.2 Finding the nth term Writing the RULE for a Linear Sequence Homework: lesson 2.2/1-8
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 5
n-3 -2 -1 0 1 2
n 1 2 3 4 5
2n+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 .. 54
6n 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 .. 37
2x+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 5
n-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 27
n 1 2 3 4 5
f(n) 9 6 3 0 -3
n 1 2 3 4 5
f(n) -8 -4 0 4 8
n 1 2 3 4 5
f(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
1a
na n represents the term you are calculating
1st term in the sequence
d the common difference between the terms