Given that the population is one-tenth as large as the prior year and that the starting population is 100,000, we have the sequence:
a_ 1 = 100000
a_ 2 = 10000
a_ 3 = 1000
a_ 4 = 100
a_ 5 = 10
Then:
[tex]\mathrm{A\:geometric\:sequence\:has\:a\:constant\:ratio\:}r\mathrm{\:and\:is\:defined\:by}\:a_n=a_1\cdot r^{n-1}[/tex]In this case, we have r = 1/10 and the equation that models this situation is:
[tex]a_n=100000\left(\frac{1}{10}\right)^{n-1}[/tex]After 5 years, the population will be:
[tex]\:a_5=100000\left(\frac{1}{10}\right)^{5-1}=10\text{ }[/tex]