|multiply both sides by -1.|
30, 31, 32, 33Edit
Although these can be done by hand the exercises suggest that you derive a general formula for finding the final population figures. The following one will suffice for the stated problems, where is the initial population, is the scaling factor (doubles, triples, etc.), and is the inital year and end year respectively, and is the number of years taken for the population to go up by :
For example, 32a:
Write out the powers of 2 not exceeding the largest number in the problem set. Systematically check to see if each power divides the given number, starting with the largest.
Write out the powers of 3 not exceeding the largest number in the problem set. Systematically check to see if each power divides the given number, starting with the largest.
Examining the definitions first, if and then and .
means that .
Examine the definitions in part a) again. We have and .
Solving for and :
120, 720, 5040 and 40320.
Mutliply both sides of the equation by and , cancelling unecessary factors to 1. We do this in order to achieve the denominator :
Use the fact that and to achieve:
Recall from the chapter that if , then .