Question

sum of geometric series (context)
score: 0/2 penalty: 1 off
question
a major fishing company does its fishing in a local lake. the first year of the companys operations it managed to catch 200,000 fish. due to population decreases, the number of fish the company was able to catch decreased by 6% each year. how many total fish did the company catch over the first 11 years, to the nearest whole number?
answer attempt 1 out of 2

Explanation:

Step1: Identify the geometric series parameters

The first term \( a = 200000 \), the common ratio \( r = 1 - 0.06 = 0.94 \) (since it decreases by 6% each year), and the number of terms \( n = 11 \).

Step2: Use the sum formula for a geometric series

The formula for the sum of the first \( n \) terms of a geometric series is \( S_n=\frac{a(1 - r^n)}{1 - r} \) (when \( r
eq1 \)).

Substitute \( a = 200000 \), \( r = 0.94 \), and \( n = 11 \) into the formula:

\( S_{11}=\frac{200000(1 - 0.94^{11})}{1 - 0.94} \)

First, calculate \( 0.94^{11} \). Using a calculator, \( 0.94^{11}\approx0.540268 \)

Then, \( 1 - 0.94^{11}\approx1 - 0.540268 = 0.459732 \)

Next, \( 1 - 0.94 = 0.06 \)

Now, \( \frac{200000\times0.459732}{0.06}=\frac{91946.4}{0.06}=1532440 \) (rounded to the nearest whole number)

Answer:

1532440