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

# Question Video: Finding the General Term of an Arithmetic Sequence in a Real-World Context Mathematics • Second Year of Secondary School

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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 sub ๐, itโs equal to ๐ plus ๐ minus one ๐. And thatโs where ๐ is our term number, ๐ 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 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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