10/31/12 Recognizing patterns Describe the growth pattern of this sequence of blocks. Illustrate the sequences of blocks for 4, 5, and 6. Try creating an equation from this sequence. 1 2 3 4 ?

Warm Up Answers Description of the growth pattern: For each term we are adding a column of 4 blocks. For a verbal explanation: http://www.khanacademy.org/math/algebra/solving-linear- equations-and-inequalities/v/equations-of-sequence-patterns Equation 1 + (x-1)4 1 + 4x -4 4x - 3

e-Learning Introduction to Sequences and Patterns http://e-learningforkids.org/Courses/EN/M1107/index.html

Sequences and Series EQ: Is a sequence a function? MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.(Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.)

Goals: Students will recognize sequences Students will identify a common differences Students will learn how to write explicit and recursive formulas for arithmetic sequences. Students will learn how to determine the number of terms in an arithmetic sequence.

Cornell Notes SEQUENCES A sequence is a function whose domain in the set of positive integers. So if I gave you a function but limited the domain to the set of positive integers and you substituted them in beginning with 1 then 2 then 3 etc. you would generate a sequence. This is a sequence generated by putting 1, 2, 3, 4, 5 … in the function above.

Identify the pattern in this sequence. 11 5 -1 -7 -13 -19 When you subtract the previous term from the current term you get a term called the common difference: d = an – an-1 d = 5 – 11 = -6 -1 + -6 = -7 Add the common difference to get the next term in the sequence.

Arithmetic Sequences – Identify the Patterns 0 3 6 9 12 +3 +3 +3 +3 10 5 0 -5 -10 -5 -5 -5 -5 +3, -5 called Common Difference ALL arithmetic sequences have a common difference.

Example What is the common difference for the sequence: 3 7 11 15 19… d = 7 -3 = 4, 11-7 = 4 Try at least twice! Common Difference = 4 What is the next term in the sequence? 19 + 4 = 23

Recursive Formula for Arithmetic Sequences The Recursive formula uses the previous term in the sequence. an = an-1 + d Example: 7 11 15 19 +4 +4 +4 +4 d = 4 Substitute 4 into your formula: an = an-1 + 4

Let’s view our example in a Table Value n a1 3 1 3 + 0 3 + 0(4) a2 7 2 3 + 4 3 + 1(4) a3 11 3 + 4 + 4 3 + 2(4) a4 15 4 3 + 4 + 4 + 4 3 + 3(4) a5 19 5 3 + 4 + 4 + 4 + 4 3 + 4(4) a1 + (n-1)d Explicit formula for Arithmetic sequence

Explicit Formula tells you how to find any value in a sequence. For any Arithmetic Sequence you can create an explicit formula using the first term and the common difference: an = a1 + (n-1)d Where a1 is the first term, n is the term number and d is the common difference.

Use the Explicit Formula to find the nth term. For a sequence with a1 = 3 and d = 4: Substitute given terms into the equation an = 3 + (n-1)4 Find the 10th term. A10 = 3 + (9)4 = 39

Find the Explicit Formula for: 6 8 10 12 14 d = 2, a1 = 6 an = 6 + (n-1)2 Can you simplify this formula? 4 + 2n

Write an explicit and recursive formula for the given sequence 5 3 1 -1 Find d. Explicit Formula: an = a1 + (n-1)d an = 5 + (n-1)(-2) = 5 – 2n + 2 = 7 – 2n You can create a Recursive Formula using the common difference: an = an-1 + d an = an-1 - 2

Find the Explicit & Recursive formula for: 3 6 9 12 15 18 an = a1 + (n-1)d an= 3 + (n-1)(3) = 3 + 3n -3 = 3n an = an-1 + d = an-1 + 3

How to determine the number of terms in an arithmetic sequence 1 , 5 , 9 , 13 ,…, 461 1. Find the Common Difference: 2. Use the explicit formula for arithmetic sequences an = a1 + (n-1)d, Where: an = last term (in this case 461) a1 = first term (in this case 1) d = common difference (in this case 4) Solve by substituting what is given into your equation. Try it! Solution: There are 116 terms in this sequence

Find the number of terms in the following sequence: 45, 32, 19, 6, …, -137 a1 = ? an = ? End of Cornell Notes

Closing – Exit Ticket Students will complete questions 9 and 10 of the Functioning Well Task. In these questions students are asked to write an explicit and recursive formula to represent graphs. Please submit task to your teacher as an exit ticket.