Exercise 101 Page 232

a To find the missing entries, we need to determine if this is an arithmetic or a geometric sequence. Let's do a rate of change table and check the difference between consecutive terms

The rate of change is not constant and therefore, the sequence is not arithmetic. Next, we will test if the change between consecutive numbers is exponential.

The population changes with a common factor of

4

from one month to the next. With this information, we can write an equation describing the geometric sequence with an initial value of

2

and a common factor of

4 .

y = 2 (4)^{x}

Now we can find the missing entries by substituting the different values of

x

in our rule.

x 3456 2 (4)^{x} 2 (4)^{3} 2 (4)^{4} 2 (4)^{5} 2 (4)^{6} y 12851220488192

Finally, we can complete the table!

Month (x) Population (y) 0218232312845125204868192

b Again, we will begin with determining if the sequence is arithmetic or geometric.

As we can see, the rate of change is not constant and therefore, the sequence is not arithmetic. Next, we will test if the rate of change is exponential.