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 find out how many years it takes for the insect population to double, we start with the equation:

$P_{n+1} = kP_n$

If we want the population to double, we set:

$P_{n+1} = 2P_n$

This gives us the equation:

$2P_n = kP_n$

Dividing both sides by $P_n$ (assuming $P_n eq 0$) leads to:

$2 = k$

Substituting the given value of $k = 1.13$ into the equation, we can look at the long-term behavior of the population increase:

To find out how long it takes to double, we can use logarithmic formulas. We know:

$P_n = P_0 k^n$

Setting this equal to $2P_0$ for the doubling condition:

$P_0 k^n = 2P_0$

Dividing by $P_0$ gives:

$k^n = 2$

Taking the logarithm of both sides gives:

$n imes ext{log}(k) = ext{log}(2)$

Thus, we can solve for $n$:

n = rac{ ext{log}(2)}{ ext{log}(k)}

Substituting $k = 1.13$ into the formula yields:

n imes ext{log}(1.13) ackslashtext{ approximately } 0.3010

Calculating this gives approximately $n = 4.09$. Therefore, it takes about 4.1 years for the insect population to double.