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Agenda 1/8 & 1/9 • • • • • • Warm up What’s My Pattern? Arithmetic Sequences Function Formula Explicit / Closed Form Homework Warm UP A square table seats 4 people. Two square tables pushed together seat 6 people. Three tables pushed together seat 8 people. How many tables are needed to seat 22 people? What’s My Pattern? Math 8 Unit 4 Task Work with your partner to complete questions number 1. What’s My Pattern? How did you do??? a) 6,7,8 b) 17, 19, 21 c) -5, -8, -11 d) 22, 29, 37 e) 4, -4, 5 f) 36, 49, 64 g) 11, 12.5, 14 h) -4, -8.3, -12.6 i) 95, 191, 383 j) 13, 21, 34 Now do question 2. What’s My Pattern? A sequence is an ordered list of numbers. Each number in the sequence is called a term. The first term,7, of the second sequence, (sequence b), is labeled as b1. You read it as “b sub one”. The second term,9, is labeled b2, the third, 11, is b3, and so on. Now do question 3. What’s My Pattern? Here is one to try on your own: k) 2, 4, 6, 8, 10, 12, 14, 16, ___, ___, ___ 2 is the 1st term (k1), 10 is the 5th term (k5), and 16 is the ____ term. What is the 10th term (k10)? __________ Arithmetic Sequences An arithmetic sequence is a sequence in which each term is equal to the previous term plus a constant. The constant is called the common or constant difference. We use the letter d to represent the constant difference. We use the letter n to tell us which term number we want. There are several ways to write a “formula” for these sequences. Arithmetic Sequences Why would we want to write a formula? How would you find the 6th term of the following sequence? 8,15,22,29,… What about the 10th term? What about the 20th term? How about the 50th term, or the 100th term? There comes a time when a formula would be helpful!! Function Formula First formula…..Function Formula. (BTW, these are only for ARITHMETIC SEQ’s!!) 1) Formula: y = dn + _?_ (I call this the adjuster) 2) Determine what is happening to your pattern. Find the constant difference,d. 3) Pick a term in the sequence. You will get 2 pieces of information from this term: y is the actual value and n is the term number. Function Formula 4) Sub d, n and y into y=dn+ ___ and find out what number needs to fill in the blank. (d is ALWAYS in front of n!) 5) You now have what you need to write your formula: d and the number that goes in the blank (adjuster) Try the following sequences: a) 8, 15, 22, 29, …. Function Formula b) 100, 93, 86, 79, … c) 2, 5, 8, 11, …. d) 11, 7, 3, -1, … Function Formula Once you have the formula you can find any term, the 15th, the 35th, the 100th, etc… Let’s use the function formula from sequence a: y =__________ a) Find the 15th term Function Formula b) Find the 30th term. c) Find the 50th term. Explicit / Closed Form 2nd Formula: Explicit / Closed Form and the Recursive Definition The Explicit / Closed form is written almost the same way. Everything is the same, EXCEPT, instead of y we use tn. 1) Formula: tn = dn + _?_ (I call this the adjuster) Explicit / Closed Form 2) Determine what is happening to your pattern. Find the constant difference, d. 3) Pick a term in the sequence. You will get 2 pieces of information from this term: tn is the actual value and n is the term number. 4) Sub d, n and tn into tn =dn+ ___ and find out what number needs to fill in the blank. (d is ALWAYS in front of n!) 5) You now have what you need to write your formula: d and the number that goes in the blank (adjuster) Explicit / Closed Form Try the following sequences: a) 8, 15, 22, 29, …. b) 100, 93, 86, 79, … c) 2, 5, 8, 11, …. d) 11, 7, 3, -1, … Explicit / Closed Form Once you have the formula you can find any term, the 15th, the 35th, the 100th, etc… Let’s use the function formula from sequence c: tn =__________ a) Find the 15th term Explicit / Closed Form b) Find the 30th term. c) Find the 50th term. Arithmetic Sequences We will work on practicing these more next class. Homework • Worktext p.421 and 422, all