The population of a city was five-sevenths of a million in 2010 and five million in 2016. The population growth can be described as an arithmetic sequence. Find its constant difference, which represents the annual growth of the population.
Well, we’re told that this represents an arithmetic sequence. Well, arithmetic sequence is a sequence that has a common difference between each term. For example, if we had two, seven, 12, 17, then we can see that this has a common difference of five because we’re adding five each time to get to the next term. And what we have in our general form is a form that says that 𝑎 sub 𝑛, so any term, is equal to 𝑎 sub one, our first term, plus 𝑛 minus one multiplied by 𝑑, where 𝑛 is the term number and 𝑑 is our common difference.
Well, in our problem, we can see that the first term is gonna be five-sevenths of a million. Well, I’ve written five-sevenths of a million, but I can actually remove the million. And that’s because we can see that we’re gonna talk about five million being the population in 2016. And as they both are million, we can just deal with the five-sevenths and the five. And we can say that 𝑎 sub seven or the seventh term is gonna be equal to five. And that’s because this is the population in 2016. And we can see that would be the seventh term in our sequence cause we’d have 2010, 2011, 12, 13, 14, 15, and then 16.
Our 𝑛 would be seven. And then 𝑑 our common or constant difference is what we’re looking for. Okay, great, so now, let’s put this into our general form. So when I do that, I’m gonna get five equals five over seven plus seven minus one multiplied by 𝑑. So this is gonna give us five equals five-sevenths plus six 𝑑. So then we can subtract five-sevenths from each side of the equation. So we’re gonna get five minus five-sevenths is equal to six 𝑑. So what I’ve done now is I’ve converted five into sevenths. And to do that, what I do is I multiplied five by seven. That gives us 35. So we got thirty-five sevenths minus five-sevenths equals six 𝑑.
So then we get thirty-sevenths is equal to six 𝑑. So then what we can do is divide by six throughout. And we do that because we wanna find out what one 𝑑, we wanna find out what our constant or common difference is. So when we divide by six, we get five-sevenths is equal to 𝑑. So therefore, we can say that the constant or common difference is going to be equal to five-sevenths. This means that the annual growth will be five-sevenths of a million.