The population of a city was a third of a million in 2010 and five million in 2016. The population can be described as an arithmetic sequence. Find the linear equation for the population 𝑝 in millions expressed in terms of the number of years 𝑛 given the growth is constant and where 𝑛 equals one is 2010.
So, here, we’re told in this question that we’ve got an arithmetic sequence. So, what does this mean? Well, an arithmetic sequence is a sequence where there’s a common difference between each of our terms. And if we’re looking at an arithmetic sequence, what we have is a general form for each term. And that is, if we have 𝑎 sub 𝑛, it’s equal to 𝑎 plus 𝑛 minus one 𝑑. And that’s where 𝑛 is our term number, 𝑎 is the first term, and 𝑑 is the common or constant difference.
Ok, great! So, we’ve got this general form. We know what an arithmetic sequence is. So, it’s a sequence with a common or constant difference between each term. But how can we use it to solve our problem? So then, we’re gonna take a look at the information we’ve been given. So, first of all, we’re told what the population in the first year, so 2010, is. And we know that’s first year cause it says where 𝑛 is equal to one.
So, that means that 𝑝 sub one is gonna be equal to a third. And that’s because we’re told that the population in 2010 is a third of a million. And because we’re dealing with millions throughout, we don’t need to write the million. So, we can just write 𝑝 sub one is equal to a third. It’s worth noting that this is, in fact, the same as our 𝑎 in our general form.
Then, we’re next told that the population is five million in 2016. Well, this is gonna be 𝑝 sub seven cause it’s gonna be our seventh term. So therefore, we can say that 𝑝 sub seven is gonna be equal to five. So therefore, what we can do is we can substitute in our values into the general form to find 𝑑, our common or constant difference.
So, when we do that, we get five is equal to a third plus seven minus one 𝑑. So, this is gonna give us five is equal to a third plus six 𝑑. So then, next, what we’re gonna do is subtract a third from each side of the equation. So, to do this, what I’m gonna do is I’m gonna convert five into thirds. So, five is fifteen-thirds. So therefore, if five is fifteen-thirds and we subtract one-third, we’re then left with fourteen-thirds. So, we got fourteen-thirds is equal to six 𝑑.
So then, if we divide through by six, we’re gonna get 14 over 18 is equal to 𝑑. And if we think about how that worked, well, if we’re doing 14 over three divided by six, it’s the same as 14 over three multiplied by one over six, which is gonna give us 14 over 18. And then, if we simplify this, we can say that 𝑑 is equal to seven-ninths, or seven over nine.
So therefore, if we put this all together to try and find out what 𝑝 — and we’re gonna call 𝑝 the population at any given year — this is gonna be equal to a third plus 𝑛 minus one multiplied by seven-ninths. Which is gonna give us 𝑝 is equal to third plus seven-ninths 𝑛 minus seven-ninths.
And then, if we simplify this, we’re gonna get 𝑝 is equal to seven-ninths 𝑛 minus four-ninths. And that’s because if we have a third minus seven-ninths, well, a third is the same as three-ninths. And three-ninths minus seven-ninths gives us negative four-ninths. And then, if we take a ninth out as a factor, we’re gonna get 𝑝 is equal to a ninth multiplied by seven 𝑛 minus four. And this is the linear equation for the population 𝑝 in millions.