Question

A TV manufacturer produces 500 TVs in its first year and plans to increase production by 8 percent each year for the next 10 years. How many total TVs will have been produced at the end of the tenth year? 6,856, 8,949, 5,687 or 7,243

Answer 1

To calculate the total number of TVs produced over 10 years with an initial production of 500 TVs and an annual increase of 8%, we can use the formula for the sum of a geometric series.

1. **Identify the initial production and growth rate**:
- Initial production (year 1, \( a \)) = 500 TVs
- Growth rate (common ratio, \( r \)) = 1 + 0.08 = 1.08
- Number of years (\( n \)) = 10

2. **Formula for the sum of a geometric series**:
\[
S_n = a \frac{(1 - r^n)}{(1 - r)}
\]
where \( S_n \) is the total production over \( n \) years.

3. **Plug in the values**:
\[
S_{10} = 500 \frac{(1 - (1.08)^{10})}{(1 - 1.08)}
\]

4. **Calculate \( (1.08)^{10} \)**:
\[
(1.08)^{10} \approx 2.1589
\]

5. **Substitute this value back into the equation**:
\[
S_{10} = 500 \frac{(1 - 2.1589)}{(1 - 1.08)}
\]
\[
S_{10} = 500 \frac{(-1.1589)}{-0.08}
\]
\[
S_{10} = 500 \cdot \frac{1.1589}{0.08}
\]
\[
S_{10} = 500 \cdot 14.48625 \approx 7243.125
\]

Since we want the total TVs produced, we round this to the nearest whole number:
\[
S_{10} \approx 7243
\]

Thus, the total number of TVs produced at the end of the tenth year is **7,243**.