4.2B Geometric Explicit and Recursive Sequences How to write recursive and explicit formulas?

Warm-Up Write an explicit equation for the following arithmetic sequence. +3

Explicit Notation - Geometric Essential Questions: How are geometric sequences written as an explicit formula and a recursive formula? Explicit Definition (review): An explicit formula allows direct computation of any term for a sequence a1, a2, a3, . . . , an, . . . .   Recursive Definition (review): Recursive formula is a formula that is used to determine the next term of a sequence using one or more of the preceding terms.

Developing the Explicit Formula for an Geometric Sequence 2, 6, 18, 54, ... Explicit Formula a1  =  2             a2  =  2(3) a3   = 2(3)(3) a4  =  2(3)(3)(3) an   =   =  2              =  2(3)¹   =  2(3)²  =  2(3)³ = 3 r = common ratio 2

Example 1a: Write an Explicit Equation

Example 1b: Use the Explicit Equation

Common Misunderstanding

Write the explicit equation for the sequence Example 2a: Write the explicit equation for the sequence {1 , 2, 4, 8 16…} x2

Example 2b:

Sequence Notation The notation on the on the second table gives us information about the order of the sequence and the position of the number.  

Developing the Recursive Formula for an Geometric Sequence 2, 6, 18, 54,... Recursive Formula Geometric Sequence Recursive Formulas have two parts The starting value of a1. The recursive equation for an as a function of an-1 (previous term) a1  =  2             a2  =  2(3) a3   = 6(3) a4  =  18(3) an   = (an-1)(3) an   = r (an-1 ) previous term previous term Common ratio Common ratio

Example 3 1, 2, 4, 8, 16, 32… a1 = ___ 2(an-1 ) an = _____ Find the recursive equation for the following geometric sequence. 1, 2, 4, 8, 16, 32… 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   = ___ 1 2(an-1 ) an   = _____

Example 4 Determine whether each situation represents an arithmetic or geometric sequence and then find the recursive and explicit equation for each. 3, 6, 9,12,... 3, 9, 27, 81,... Arithmetic or Geometric Common Difference:_______________ or Common Ratio:___________________ Arithmetic or Geometric       Recursive:____________________ Explicit:_______________________ Recursive:_____________________ Explicit:___________________ d = 3 none r = 3 none a1 = 3,  an = an-1+3 a1 = 3,  an = 3(an-1) an = 3+3(n-1)  or an = 3n   an = 3(3)n-1 an = 3n

Example 5: The table below represents an arithmetic sequence. Find the missing terms of the sequence, showing your method. Find the total number of differences. 2, _____  ,  _____, 26 x y 1 2 ??? 3 ???   4 26 Total difference between term 1 and term 4 is 24. 10 How many additions must be performed? 18

Example 6 Fill in the table below for day 5 and 6. Create a possible scenario for the table provided below. Remember to graph and label. Write a recursive equation.

Example 7