A competitor is running a 20 kilometre race - Edexcel - A-Level Maths Pure - Question 11 - 2019 - Paper 2

The rth kilometre can be understood based on the time taken for the first 4 kilometres and how each subsequent kilometre takes 5% longer.

For r = 5, the time for the 5th kilometre equals:

$6 \text{ minutes} \times 1.05^{5-4} = 6 \times 1.05^1$

For r = 6, it is:

$6 \text{ minutes} \times 1.05^{6-4} = 6 \times 1.05^2$

Continuing this pattern gives us the formula:

$\text{Time for rth km} = 6 \times 1.05^{r-4}$

for 5 ≤ r < 20.

To find the total time for the entire 20 km, we start from the first 6 km:

We have already calculated:

$\text{Total time for first 6 km} = 36.915 \text{ minutes}$

Next, for the kilometres from 7 to 20, these form a geometric series. The time for the nth term is:

$T_n = 6 \times 1.05^{n-4}$

To find the total time from 7 to 20 km:

$\text{Total from } 7 \text{ to } 20 = \sum_{n=7}^{20} 6 \times 1.05^{n-4}$

This is a geometric series where:

• First term (a) = 6 × 1.05^3
• Common ratio (r) = 1.05
• Number of terms (n) = 14 (from 7 to 20 inclusive)

Using the formula for the sum of a geometric series:

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

Plugging in the values:

$S_{14} = 6 \times 1.05^3 \times \frac{1 - 1.05^{14}}{1 - 1.05}$

Calculating this yields approximately: 149.027 minutes.

Adding the first 6 km gives the total:

Total time = 36.915 + 149.027 ≈ 185.942 minutes

Converting this to minutes and seconds results in approximately 173 minutes and 3 seconds.