Question Video: Finding the General Term of an Arithmetic Sequence in a Real-World Context Mathematics

The population of a city was 1/3 of a million in 2010 and 5 million in 2016. The population can be described as an arithmetic sequence. Find the linear equation for the population ๐‘ in millions expressed in terms of the number of years ๐‘› given the growth is constant and where ๐‘› = 1 is 2010.

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Video Transcript

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 ๐‘ in millions expressed in terms of the number of years ๐‘› given the growth is constant and where ๐‘› equals one is 2010.

So, here, weโ€™re told in this question that weโ€™ve got an arithmetic sequence. So, what does this mean? Well, an arithmetic sequence is a sequence where thereโ€™s a common difference between each of our terms. And if weโ€™re looking at an arithmetic sequence, what we have is a general form for each term. And that is, if we have ๐‘Ž sub ๐‘›, itโ€™s equal to ๐‘Ž plus ๐‘› minus one ๐‘‘. And thatโ€™s where ๐‘› is our term number, ๐‘Ž is the first term, and ๐‘‘ is the common or constant difference.

Ok, great! So, weโ€™ve got this general form. We know what an arithmetic sequence is. So, itโ€™s a sequence with a common or constant difference between each term. But how can we use it to solve our problem? So then, weโ€™re gonna take a look at the information weโ€™ve been given. So, first of all, weโ€™re told what the population in the first year, so 2010, is. And we know thatโ€™s first year cause it says where ๐‘› is equal to one.

So, that means that ๐‘ sub one is gonna be equal to a third. And thatโ€™s because weโ€™re told that the population in 2010 is a third of a million. And because weโ€™re dealing with millions throughout, we donโ€™t need to write the million. So, we can just write ๐‘ sub one is equal to a third. Itโ€™s worth noting that this is, in fact, the same as our ๐‘Ž in our general form.

Then, weโ€™re next told that the population is five million in 2016. Well, this is gonna be ๐‘ sub seven cause itโ€™s gonna be our seventh term. So therefore, we can say that ๐‘ 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 ๐‘‘, our common or constant difference.

So, when we do that, we get five is equal to a third plus seven minus one ๐‘‘. So, this is gonna give us five is equal to a third plus six ๐‘‘. So then, next, what weโ€™re gonna do is subtract a third from each side of the equation. So, to do this, what Iโ€™m gonna do is Iโ€™m gonna convert five into thirds. So, five is fifteen-thirds. So therefore, if five is fifteen-thirds and we subtract one-third, weโ€™re then left with fourteen-thirds. So, we got fourteen-thirds is equal to six ๐‘‘.

So then, if we divide through by six, weโ€™re gonna get 14 over 18 is equal to ๐‘‘. And if we think about how that worked, well, if weโ€™re doing 14 over three divided by six, itโ€™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 ๐‘‘ is equal to seven-ninths, or seven over nine.

So therefore, if we put this all together to try and find out what ๐‘ โ€” and weโ€™re gonna call ๐‘ the population at any given year โ€” this is gonna be equal to a third plus ๐‘› minus one multiplied by seven-ninths. Which is gonna give us ๐‘ is equal to third plus seven-ninths ๐‘› minus seven-ninths.

And then, if we simplify this, weโ€™re gonna get ๐‘ is equal to seven-ninths ๐‘› minus four-ninths. And thatโ€™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, weโ€™re gonna get ๐‘ is equal to a ninth multiplied by seven ๐‘› minus four. And this is the linear equation for the population ๐‘ in millions.

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