A trading company made a profit of £50 000 in 2006 (Year 1) - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 2

To show that the profit exceeds £200 000:

[ 50000r^{n-1} > 200000 ]

Dividing both sides by £50 000 gives:

[ r^{n-1} > 4 ]

Taking logarithms of both sides:

[ (n-1) \log r > \log 4 ]

Solving for n:

[ n-1 > \frac{\log 4}{\log r} ]

Thus,

[ n > \frac{\log 4}{\log r} + 1 ]

Using r = 1.09, we can calculate:

[ n > \frac{\log 4}{\log 1.09} + 1 ]

Calculating the logs:

• (\log 4 \approx 0.60206)
• (\log 1.09 \approx 0.03743)

Substituting these values in:

[ n > \frac{0.60206}{0.03743} + 1 \approx 17.0 ]

Therefore, the profit will first exceed £200 000 in Year 18 (2023).

To find the total profits over 10 years, we can use the formula for the sum of a geometric series:

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

Where:

• ( a = 50000 )
• ( r = 1.09 )
• ( n = 10 )

Calculating:

• ( S_{10} = 50000 \frac{1 - 1.09^{10}}{1 - 1.09} )
• ( 1.09^{10} \approx 2.36736 )

Then: [ S_{10} = 50000 \frac{1 - 2.36736}{-0.09} = 50000 \frac{-1.36736}{-0.09} = 760000 ]

Thus, the total profit over the 10 years is approximately £760 000.