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Math I Day 10 (1-19-10) Today’s Question: How can we write a function to represent a sequence? Standard: MM1A1f: Recognize sequences as functions with domains that are whole numbers Sequences 0011 0010 1010 1101 0001 0100 1011 • Everyday a radio station asks a question for a prize of $150. If the fifth caller does not answer correctly, the prize money increased by $150 each day until someone correctly answers their question. • Make a list of the prize money for a week from Monday to Friday if no one gets it right. 1 2 4 Contest 0011 0010 1010 1101 0001 0100 1011 Monday: $150 Tuesday: $300 Wednesday: $450 Thursday: $600 Friday: $750 1 2 4 Sequences 0011 0010 1010 1101 0001 0100 1011 These prize amounts form a sequence. Sequence: A function whose domain is a set of consecutive whole numbers. 1 2 4 What would be the value on Day 6? Day 7? An Arithmetic Sequence: is a sequence that has a common number added to every term Sequences The values in the range are called the terms 0011 0010 1010 1101 0001 0100 1011 of the sequence. Domain: 1 2 3 4…....n Range: a2 a3 a4….. an a1 1 2 4 A sequence can be specified by an equation or rule. Contest 0011 0010 1010 1101 0001 0100 1011 a1 $150 a2 $300 a3 $450 a4 $600 a5 $750 1 2 4 an represents a general term where n can be any number. What is the closed formula for an? Sequences 0011 0010 1010 1101 0001 0100 1011 Sequences can continue forever. We can calculate as many terms as we want as long as we know the rule or equation for an. Example: 3, 5, 7, 9, ___ , ___,……. _____ . an = 2n + 1 1 2 4 Writing Terms of a Sequence 0011 0010 1010 1101 0001 0100 1011 an = 2n – 3 Find a1, a2, a3, a4, a5 a1 = -1 First Term a2 = 1 Second Term a3 = 3 Third Term a4 = 5 Fourth Term a5 = 7 Fifth Term 1 2 4 Example 1 0011 0010 1010 1101 0001 0100 1011 an = 2n – 20 Find a1, a2, a3, a4, a5 a1 = -18 First Term a2 = -16 Second Term a3 = -14 Third Term a4 = -12 Fourth Term a5 = -10 Fifth Term 1 2 4 Recursive rule v. Closed rule 0011 0010 1010 1101 0001 0100 1011 • Recursive rule: relates one term to the previous term – How to find:state the first term (t1=?), then state what operation much occur on that term to get to the next term (tn=tn-1+d) 1 2 4 • Closed rule: allows you to calculate any term in the sequence directly – How to find: the equation will always have the form tn=dn+b, where d is the number you added in the recursive rule, and b is whatever number will make tn=t1 when n=1 Example 2 0011 0010 1010 1101 0001 0100 1011 a1 = 2 First Term a2 = 6 Second Term a3 = 10 Third Term a4 = 14 Fourth Term an = nth Term What is the recursive rule? 1 4 t1 = ???, tn-1 = ??? t1= 2, tn=tn-1+4 What is the closed rule? tn = ??? tn= 4n - 2 2 Example 3 0011 0010 1010 1101 0001 0100 1011 a1 = 5 First Term a2 = 12 Second Term a3 = 19 Third Term a4 = 26 Fourth Term an = nth Term 2 4 What is the recursive rule? t1 = ???, tn-1 = ??? t1=5, tn=tn-1+7 What is the closed rule? tn = ??? tn= 7n - 2 1 EOCT Practice a) 771101 0001 0100 1011 0011 0010 1010 c) 84 b) 81 d) 86 1 2 4 b EOCT Practice Find the 25th term of the sequence 7, 11, 15, 19, 23, ... 0011 0010 1010 1101 0001 0100 1011 a) 103 c) 107 b) 104 d) 111 1 2 4 a