Question

question this year, the number of new saplings in a deer exclosure is 682. it is estimated that the number of new saplings will decrease by 9% each year. find the total number of saplings at the end of 9 years. (round your answer to the nearest whole number if necessary.) provide your answer below:

Explanation:

Step1: Identify the formula for a geometric - series

The number of new saplings forms a geometric series with first - term \(a = 682\) and common ratio \(r=1 - 0.09=0.91\). The sum of a geometric series \(S_n=\frac{a(1 - r^n)}{1 - r}\) for \(n\) terms.

Step2: Substitute the values into the formula

Here, \(a = 682\), \(r = 0.91\), and \(n = 9\). So \(S_9=\frac{682\times(1 - 0.91^{9})}{1 - 0.91}\).
First, calculate \(0.91^{9}\):
\(0.91^{9}=0.91\times0.91\times\cdots\times0.91\) (9 times). Using a calculator, \(0.91^{9}\approx0.40158\).
Then, \(1 - 0.91^{9}=1 - 0.40158 = 0.59842\).
Next, \(682\times(1 - 0.91^{9})=682\times0.59842\approx407.1224\).
Finally, \(\frac{682\times(1 - 0.91^{9})}{1 - 0.91}=\frac{407.1224}{0.09}\approx4523.58\).

Answer:

4524