Video: Finding the Sum of Arithmetic Sequences

Find the value of (7 + 9 + 11 + _ + 37)/(12 + 14 + 16 + _ + 172).


Video Transcript

Find the value of seven plus nine [plus 11] plus dot dot dot plus 37 divided by 12 plus 14 plus 16 plus dot dot dot plus 172.

So those dot dot dots means that there are numbers in between there. These are sequences on the numerator and the denominator. Both the numerator and the denominator are arithmetic sequences. We know this because to get the next term in the sequence, we’re adding a constant number to each term to get the following term. And we call that term that’s being added a common difference which we will use as 𝑑 to represent that. And then our first term is important too for our formula, and that’s 𝑎.

And now we have a formula for the numerator and denominator to find out how much they’re actually worth if we would add all the numbers together, even though we don’t know every single number, at least that is not given to us. 𝑆 sub 𝑛 is called the partial sum. And it’s the partial sum because we know that this sum on the numerator and denominator does end; it’s not continuous. It doesn’t go on forever.

So let’s first begin by looking at our numerator. Our first term is seven, so that would be 𝑎. Our common difference between each term would be two because to get from seven to nine we add two, to get from nine to 11 we add two. And now the only thing that’s left is 𝑛, and there’s a formula to find 𝑛. 𝑎 sub 𝑛 equals 𝑎 sub one plus 𝑛 minus one times 𝑑. 𝑎 sub 𝑛 would be our last term and 𝑎 sub one would be our first term. And it’s also given that this is an arithmetic sequence, so this is what we would use for arithmetic sequences.

So let’s first look at our numerator to find this 𝑛th term. Our last term is 37, our first term is seven, and our common difference is two. So now let’s simplify. Let’s subtract seven from both sides and distribute the two. Now let’s add two to both sides and now divide both sides by two. So 𝑛 is equal to 16. Now we have everything we need to find the sum of the numerator. We plug in 16 for 𝑛, seven for 𝑎, and two for 𝑑. Two times seven is 14 and 16 minus one is 15, but times two is 30. So now let’s add 30 and 14 together and then multiply by eight. We get 352.

Now let’s do the exact same thing for the denominator. Our first term is 12. Our common difference is two again because 12 plus two is 14, 14 plus two is 16, and now we need to solve for 𝑛. 172 is our last term, 12 is our first term, and then our common difference will be two. So let’s go ahead and subtract 12 from both sides and distribute the two. Now let’s add two to both sides and now divide by two to both sides. So 𝑛 is equal to 81.

Now let’s plug these in to find the sum of the denominator. So we plugged in 81 for 𝑛, 12 for 𝑎, and two for 𝑑. 81 divided by two is 40.5, two times 12 is 24, and then 81 minus one is 80 but times two is 160. So now let’s add 160 and 24 and then multiply by 40.5. And we get 7452. So our numerator added to be 352 and our denominator added to be 7452. So let’s reduce, and this will be our final answer which is 88 divided by 1863.

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