The population of a town is 20000 at the start of the year 2018. A population model predicts this population will grow by 2% each year. (a) Find the estimated population at the start of 2022.
The first piece of information we need to extract from the question is the area of maths we are working in. The populations at the start of each year form a sequence. The first term of the sequence is 20,000, corresponding to n=1 and the year 2018. The second term of the sequence is 20400, which is a 2% increase from the previous year, and corresponds to n=2 and the year 2019. We can go on like this, working out the population at the start of each year by taking a 2% increase from the previous year. As you can see, this is not an arithmetic sequence, because we are not adding on an amount each time. Instead, it is a geometric sequence. Although this might not have been clear from the question, a 2% increase each time is equivalent to a geometric sequence with a common ratio of 1.02. This is because multiplying the previous term by 1.02 is the same operation as increasing by 2%.
So, we have a geomtric sequence. The r value is 1.02, and the a value is 20,000. So the general formula for a term is then Un = ar^(n-1), ie Un = 20,0001.02^(n-1). The first term of the sequence is n=1, corresponding to the year 2018. Therefore, the year 2022 corresponds to the 5th term in the sequence. U5 = 20,000*(1.02)^4, so U5 = 21648.643...
