3.5 Arithmetic Sequences as Linear Functions Objective: 1) Use inductive reasoning in continuing number patterns 2) Write rules for arithmetic sequences 3) Relate arithmetic sequences to linear functions

Vocabulary Inductive reasoning: Making conclusions based on patterns you observe. Sequence: a set of numbers in a specific order Terms: the numbers in a sequence

Vocabulary Continued Arithmetic Sequence: A numerical pattern that increase or decrease at a constant rate or value. Common Difference: The difference between the terms in a sequence. Represented by the letter d.

Find the next two numbers in each pattern and describe the pattern 2, 5, 8, 11,… 2, 4, 6, 8,… 1, 9, 17, 25, … Pattern is add by 3, next two terms are 14, and 17 Add 2, 10, 12 Add 8, 33, and 41

Determine whether the sequence is an arithmetic sequence if so what is the pattern? -4, -2, 0, 2,… Yes the pattern is add 2

Determine whether the sequence is an arithmetic sequence if so what is the pattern? -26, -22, -18, -14,… Yes, the pattern is add 4

Determine whether the sequence is an arithmetic sequence if so what is the pattern? 1, 4, 9, 25, …

Determine whether the sequence is an arithmetic sequence if so what is the pattern? The last 2 are a difference of 1/16th NO

Find the nth term a1= first term of sequence d= common difference **** n must be positive ****

Write an equation for the nth term of the arithmetic sequence -12, -8, -4, 0, … Step 1: Find the common difference Step 2: Write an equation A(n) = -12 + (n – 1)4 A(n) = -12 + 4n – 4 A(n) = - 16 + 4n

Find the 9th term of the sequence A(n) = -16 + 4n A(9) = -16 + 4(9) A(9) = -16 + 36 A(9) = 20 The 9th term is 20. STRESS TO STUDENTS THAT THE A(N) IS JUST A SUBSCRIPT AND IS NOT INVOLVED IN ANY CALCULATIONS.

Which term of the sequence is 32? A(n) = -16 + 4n 32 = -16 + 4n 48 = 4n n= 12 32 is the 12th term of the sequence

You Try Consider the arithmetic sequence 3, -10, -23, -36, … Write an equation for the nth term of the sequence Find the 15th term in the sequence Which term of the sequence is –114? a) A(n) = 3 + ( n + 1) -13 A(n) = 3 + -13n – 13 A(n) = -10 – 13n b) A(15) = -10 – 13(15) A(15) = -10 – 195 A(15) = -205 The 15th term of the sequence is -205 -114 = -10 – 13n -104 = -13n 8 = n -114 is the 8th term of the sequence

Arithmetic Sequences as Functions Marisol is mailing invitations for her quinceanera. The arithmetic sequence $0.41, $0.82, $1.23, $1.64, … represents the cost of postage. Write a function to represent this sequence A(N) = $0.41 + (n – 1) .41 A(n) = $0.41 + .41n - .41 A(n) = .41n

Write an arithmetic sequence on a blank piece of paper. 1) Have students write an arithmetic sequence on a blank piece of paper. 2) Have the students switch with the person closest to them. That person will write an equation for the sequence. 3) Have students switch with someone different. That person will find the 20th term of the sequence.