Respuesta :
The number of Fishes caught by the company over the first 8 years is 1,034,410 fishes
Here, we want to find the sum of the fishes the company was able to catch over the first 8 years
We start by setting up an exponential equation to represent the number of fishes caught at any number of years in time
We can have the exponential equation as;
[tex]P(t)=I(1-r)^{t-1}[/tex]where I is the population of fishes in the first year
r is the percentage decrease which is 8% and it is same as 8/100 = 0.08
t is the year number
and P(t) is the population at a certain year
So substituting these values, we have;
[tex]\begin{gathered} P(t)=170,000(1-0.08)^{t-1} \\ \\ P(t)=170,000(0.92)^{t-1} \end{gathered}[/tex]So in this case, we have a geometric series with the nth term given above such that;
The first term a, is 170,000
The common ratio is 0.92
So the sum of the first 8 years which is the sum of the first 8 terms can be obtained using the formula for the geometric series as follows;
[tex]\begin{gathered} S_n\text{ = }\frac{a(1-r^n)}{1-r} \\ \\ S_8\text{ = }\frac{170,000(1-0.92^8)}{1-\text{ 0.92}} \\ \\ S_8\text{ =}\frac{170,000(0.48678)}{0.08} \\ \\ S_8\text{ = }\frac{82,752.792}{0.08} \\ \\ S_8\text{ = 1,034,409.89} \\ \\ To\text{ the nearest integer, this is 1,034,410} \end{gathered}[/tex]