Lesson 2.2 Finding 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

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)

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

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

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

Writing the Rule/ nth 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 + 12 Common Difference +/- something

Finding the nth Term n +6 Find the Common Difference 1 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

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?

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

Use the pattern to find the rule & the missing term 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)+ _?__ = 6 n=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)+ _?__ = 7 n=2  2(2)+ ? =9 ? = 5 RULE: 2n+5

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

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

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

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.

IS THE PATTERN LINEAR? n 1 2 3 4 5 f(n) -3 -1 11 27 n 1 2 3 4 5 f(n) 9 6 -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

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

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