The number of insects in a population at the start of the year n is $P_n$ - Edexcel - GCSE Maths - Question 12 - 2022 - Paper 2

To determine how many years it takes for the insect population to double, we start from the equation:

$P_{n+1} = k P_n$

Given that we want the population to double:

$P_{n+1} = 2 P_n$

We can set this equal to the earlier equation:

$2 P_n = k P_n$

Dividing both sides by $P_n$ (assuming $P_n \neq 0$), we have:

$2 = k$

Since $k = 1.13$, we will find the time taken to reach the doubling by calculating:

$P_n = P_0 \times k^n$

Setting up the equation for doubling:

$2 P_0 = P_0 \times (1.13)^n$

Dividing both sides by $P_0$ gives:

$2 = (1.13)^n$

To solve for $n$, we take the logarithm:

$\log(2) = n \cdot \log(1.13)$

Thus, we can rewrite for $n$:

$n = \frac{\log(2)}{\log(1.13)}$

Using a calculator, we find:

$$\log(1.13) \approx 0.0531 \\$$

Calculating:

$n \approx \frac{0.3010}{0.0531} \approx 5.66$

Therefore, it takes approximately 6 years for the population to double.