4.2A Arithmetic Explicit and Recursive Sequences How to write recursive and explicit formulas?

Warm-Up f(x)=3x – 2 f(x)=3x-2 x f(x) 2 4 3 7 ?

Explicit Notation-Arithmetic Essential Question: How are arithmetic sequences written as an explicit formula? Explicit Definition: An explicit formula allows direct computation of any term for a sequence a1, a2, a3, . . . , an, . . . .  

Developing the Explicit Formula for an Arithmetic Sequence 1, 5, 9, 13, 17, 21,... Explicit Formula a1  =  1             a2  =  1+4 a3   = 1+4+4 a4  =  1+4+4+4 an   =   =  1              =  1+4(1)   =  1+4(2)  =  1+4(3) = 1+4(n - 1) d = common difference 4 1 n

Example 1a: Write an Explicit Equation +3

Example 1b: Use the Explicit Equation

Write the explicit equation for the sequence Example 2a Write the explicit equation for the sequence {18, 13, 8, 3,-2…} ? -5

Example 2b

Example 3 Describe the pattern. What is happening to each term to get the next? Write the explicit equation. Complete the pattern 3, 6, 9, ___,___,___,... Complete the pattern 5, 10, 15, __,__,__… 12, 15, 18, The numbers in the sequence are multiples of 3 – Inceasing by 3 each time an = 3n Explicit Equation 20, 25, 30, The number in the sequence are multiples of 5 – Adding 5 each time. an = 5n Explicit Formula

4.2A Recursive Notation - Arithmetic Essential Question: How are arithmetic sequences written as a recursive formula? Recursive Definition: Recursive formula is a formula that is used to determine the next term of a sequence using one or more of the preceding terms. The recursive sequence can be denoted by f(n) or fn  

Warm Up Using the given information. Write the next 3 terms in the sequence. The first term is 9, ___,___,___,... The first term is 1, ___,___,___… 12, 15, 18, The previous term is being increase by 3 4 , 16, 64, The previous term is being multiplied by 4

Developing the Recursive Formula for an Arithmetic Sequence 1, 5, 9, 13, 17, 21,... Recursive Formula Recursive Formulas have two parts The starting value of a1. The recursive equation for an as a function of an-1 (previous term) a1  =  1               a2  =  1+4 a3   = (1+4)+4 a4  =  (1+4+4)+4 a5   = (1+4+4+4)+4 an   =  =  1            =  1   + 4  =  5   + 4  =  9   + 4  = 13  + 4  = an-1  + 4 an   = an-1  + d previous term previous term Common difference Common difference

Why? Why are we learning how to write two different formulas for the same sequence????? https://learnzillion.com/lessons/3306-apply-recursive-and-explicit-formulas

Example 4 2, 4, 6, 8,... 2 a1 = _____ an = an-1 + d an = an-1 + 2 Find the recursive equation for the following arithmetic sequence. 2, 4, 6, 8,... 2 Remember Recursive Formulas have two parts The starting value of a1. The recursive equation for an as a function of an-1 (previous term) a1   = _____ an   = an-1  + d an   = an-1  + 2

Example 5 5, 8, 11, 14, 17,... a1 = _____ 5 an = an-1 + d Find the recursive equation for the following arithmetic sequence. 5, 8, 11, 14, 17,... a1   = _____ 5 an   = an-1  + d an   = an-1  + 3