Exponential Equations Recursive Routines

Recursive Routine Procedure that is applied over and over again, starting with a number or figure The next number of figure builds off of the previous Look for a pattern Non Linear Constant multiplier to the pattern or problem

Using the calculator {first term, first value} (ans(1)+1, Ans(2)+550} Repeated addition (ans(1)+1, Ans(2)*(1+.05}Repeated multiplication Hit return again and it will continue to do the recursive function

Investigation 1 Every day new bugs hatch increasing the population by 50% each week. Complete the table for 7 weeks

Increase in number of bugs Weeks Elapsed Total Number of Bugs Increase in number of bugs Ratio of this week's total to last weeks total Start(0) 16   1  24 8 24/16=3/2 2 36  12 36/24=3/2  3  54  18  3/2 4  81  27 5  121.5  40.5 6  182.25  60.75 7  273.375  91.125

Graph your data

Exponential Equations Repeated Multiplication In the previous problem what is the common multiplication 3/2 16*3/2=24*3/2=36*3/2=54 Is 16*(3/2)^3 the same as the above

Cont’d First value * repeated multiplication^x=new value Y=AB^x A is first value (starting value) B is constant multiplier X is what term in the sequence we are looking for Y=total or new value at that location

Example Given the sequence start the starting value, the constant multiplier, the equation, and find the 7th term in the sequence 16,20,25,31.25,… First value is 16 Common multiplier is 20/16=1.25 16(1.25)^x=y Test (remember we start with term 0 using this equation) 16(1.25)^0=16 16(1.25)^6=61.035

Example 2 Write the equation given the first term and the constant multiplier, write out the first 6 values {0,100} Constant multiplier is -1.6 100(-1.6)^x=u -160,256,-409.6,655.36,-1048.576

Exponential Growth Y=A(1+r)^x (decay is similar but with a minus R is the percent 1 is to include the original amount when increasing If minus it gives you what is left over Alice and the Deforestation