Bellwork  Extend each pattern 3 more terms, then describe in words how each term relates to the one previous. a) -3, 0, 3, 6, … b) 2, 4, 8, … c) 9, 12, 15 Each term is the previous term plus 3 16, 32, 64 Each term is the previous term multiplied by 2 Each term is the previous term multiplied by 1/2

End in Mind  Title your notes page Sequences- Day 1  Put a sub-header “End-in-mind”  Copy the problems below and use any strategies/resources to solve. A) A car whose original value was $25,000 decreases in value by $250 per month. How long will it take before the car’s value falls below $20,000? B) A car whose original value was $25,000 decreases in value by 5% per month. After 1 year, how much will the car be worth?

 Sequences: A set of values arranged in a specific order, a pattern  Recursive Process: Used to describe a pattern or sequence by describing how to get from one term to the next.  Explicit Expression: Used to describe a pattern or sequence so that any term in the sequence can be found. Vocabulary

The value added each time is called the "common difference" The common difference could also be negative: Example: 25, 23, 21, 19, 17, 15,... This common difference is −2 (Vocabulary Continued) We call the common difference ‘d’

The value multiplied each time is called the "common ratio" The common ratio could also be a fraction: Example: 48, 24, 12, 6, 3, 1.5,... This common ratio is 1/2 (Vocabulary Continued) We call the common ratio ‘r’

Use a Recursive Process to determine the 10 th y-value +4 Each y-value is the previous one plus 4. So the 10 th term is… 41 Arithmetic

Use an Explicit Expression to find the 10 th term Each y-value is the x-value times 4, plus 1. So… y=4x+1 and the 10 th term is… y = 4(10)+1 = 41 Still Arithmetic x4, +1 Remember: We added 4 each time…

To determine the Explicit Expression for Arithmetic Sequences 1) Determine what you are adding each time. (the common difference) x (term) y Added 4 each time, so we start off with y=4x… y=4x would lead to… y=4(1)=4 2) Adjust to fit the pattern Our first term is -2, NOT 4. So we need to subtract 6. y=4x-6

For each of the following arithmetic sequences. (a) Determine the 8 th term using the Recursive process. (b) Determine the 20 th term using an Explicit Expression x (term) y x (term) y x (term) y I will come around and check these as you complete them. Sequences Day 1- I.C. Practice

Use a Recursive Process to determine the 10 th y-value x3 Each y-value is the previous one times 3. So the 10 th term is… 78,732 Geometric

Use an Explicit Expression to find the 10 th term 4 4x3 4x3x3 4x3x3x3 Each y-value is the first y-value times 3 to the x-value minus 1. So… y=4 x 3 x-1 and the 10 th term is… y = 4 x =4 x 3 9 =78,732 Still Geometric 4 x 3 3 Remember: We multiplied by 3 each time…

To determine the Explicit Expression for Geometric Sequences 1) Determine what you are multiplying by each time. (the common ratio) x (term) y Multiply each time by -2, so we start off with y=(-2) x-1 y=(-2) x-1 would lead to… y=(-2) 1-1 =(-2) 0 =1 2) Adjust to fit the pattern Our first term is 2, NOT 1. So we need to multiply by 2. y=2(-2) x-1

For each of the following geometric sequences. (a) Determine the 6 th term using the Recursive process. (b) Determine the 12 th term using an Explicit Expression x (term) y 14,096 22,048 31, x (term) y x (term) y I will come around and check these as you complete them. Sequences Day 1- I.C. Practice

Ticket Out a) Come up with your own example of an arithmetic sequence and state the explicit expression that corresponds with it. b) Come up with your own example of a geometric sequence and state the explicit expression that corresponds with it.

Notation:  d= common difference (what is added to each term of an arithmetic sequence)  r= common ratio (what is multiplied to each term of a geometric sequence)  n= What term number you are looking at (4 th term, 10 th term… n th term) 1 st term2 nd term3 rd term4 th term5 th termnth term a1a1 a2a2 a3a3 a4a4 a5a5 anan