Answer:
a) The recurrence formula is .
b) The general formula for the population of Tacoma is
.
c) In 2016 the approximate population of Tacoma will be 794062 people.
d) The population of Tacoma should exceed the 400000 people by the year 2009.
Step-by-step explanation:
a) We have the population in the year 2000, which is 200 000 people. Let us write . For the population in 2001 we will use
, for the population in 2002 we will use
, and so on.
In the following year, 2001, the population grow 9% with respect to the previous year. This means that is equal to
plus 9% of the population of 2000. Notice that this can be written as
.
In 2002, we will have the population of 2001, , plus the 9% of
. This is
.
So, it is not difficult to notice that the general recurrence is
.
b) In the previous formula we only need to substitute the expression for :
.
Then,
.
Repeating the procedure for we get
.
But we can do the same operation n times, so
.
c) Recall the notation we have used:
for 2000,
for 2001,
for 2002, and so on. Then, 2016 is
. So, in order to obtain the approximate population of Tacoma in 2016 is
d) In this case we want to know when , which is equivalent to
.
Substituting the value of , we get
.
Simplifying the expression:
.
So, we need to find the value of such that the above inequality holds.
The easiest way to do this is take logarithm in both hands. Then,
.
So, .
So, the population of Tacoma should exceed the 400 000 by the year 2009.
