A TV manufacturer produces 500 TV's in its first year and plans to increase production by 8% each year for the next 10 years. How many total TVs will have been produces at the end of the tenth year?

Question

1)5,687
2)8,949
3)7,243
4)6,856

Answer 1

To calculate the total number of TVs produced over the ten years, we can use the formula for the sum of a geometric series.

The first term \( a = 500 \) (the number of TVs produced in the first year), and the common ratio \( r = 1 + 0.08 = 1.08 \) (since production increases by 8% each year).

We can calculate the number of TVs produced in each of the 10 years and sum them up. The number of TVs produced in year \( n \) can be calculated as:

\[ TVs_n = 500 \times (1.08)^{(n-1)} \]

For 10 years, we need to sum the number of TVs produced each year from \( n=1 \) to \( n=10 \):

\[ \text{Total TVs} = \sum_{n=0}^{9} 500 \times (1.08)^n \]

This is a geometric series where \( n \) goes from 0 to 9 (10 terms total), and it can be summed using the formula for the sum of a geometric series:

\[ S_n = a \frac{(1 - r^n)}{(1 - r)} \]

In our case \( a = 500 \), \( r = 1.08 \), and \( n = 10 \):

\[ S_{10} = 500 \frac{(1 - (1.08)^{10})}{(1 - 1.08)} \]

Calculating \( (1.08)^{10} \):

\[ (1.08)^{10} \approx 2.15892 \]

Now plugging into the formula:

\[ S_{10} = 500 \frac{(1 - 2.15892)}{(-0.08)} = 500 \frac{-1.15892}{-0.08} \] \[ S_{10} = 500 \times 14.4865 \approx 7243.25 \]

Rounding this, the total number of TVs produced over the 10 years is approximately 7,243.

So the answer is:

3) 7,243