Question Video: Evaluating the Sum of a Finite Series Mathematics

Video Transcript

Evaluate the sum from 𝑟 equals three to five of negative eight over 𝑟.

When we read this notation, we read it as the sum from 𝑟 equals three to five of some expression. Here, that expression is negative eight over 𝑟. And so what this question is really asking us to work out is what happens when we find the sum of negative eight over 𝑟 for values of 𝑟 between three and five inclusive. And there are two ways to answer this. Let’s consider both methods. Let’s begin by evaluating the expression negative eight over 𝑟 for 𝑟 equals three, four, and five. When 𝑟 is equal to three, the expression is negative eight-thirds. Let’s repeat this for 𝑟 equals four. This time we get negative eight over four, which is of course equal to negative two. And finally, when 𝑟 is five, we get negative eight over five. The sum of these is what we’re looking to evaluate, so we have negative eight-thirds plus negative two plus negative eight-fifths.

To find the sum of these expressions, we’ll need to create a common denominator. The lowest common multiple of three and five is 15, so let’s use that. For negative eight-thirds, we multiply both parts by five, giving us negative 40 over 15. Then, for negative two, we think about that as negative two over one. And when we multiply both parts by 15, we get negative 30 over 15. Finally, we multiply eight and five by three, and we get negative 24 over 15. So, to find the sum from 𝑟 equals three to five of negative eight over 𝑟, we add the numerators. Negative 40 plus negative 30 plus negative 24 is negative 94. So this expression becomes negative ninety-four fifteenths.

But of course we said there were two methods. The other method involves using a calculator. So we’ll double-check this solution by using our calculator. Some calculators will have the feature to allow us to type in the sum exactly as we see it. However, some we will need to perform some manipulation. If we’re finding the sum of the third, fourth, and fifth terms of the sequence where the 𝑛th term is negative eight over 𝑛, then it follows that we can find the sum of the first five terms and take away the sum of the first two. That will leave us with the sum of the third, fourth, and fifth.

Now, of course, our expression is in terms of 𝑟, not 𝑛, but otherwise the process is the same. We find the sum from 𝑟 equals one to five of negative eight over 𝑟. And then we subtract the sum from 𝑟 equals one to two of negative eight over 𝑟. Well, the sum from 𝑟 equals one to five of our expression, according to our calculator, is negative 274 over 15. And the sum from 𝑟 equals one to two is simply negative 12. It’s worth noting that we might need to replace 𝑟 with 𝑥 when typing this into our calculator, depending on the make and model. So we now need to work out negative 274 over 15 minus negative 12, which once again gives us negative 94 over 15. So we’ve evaluated the sum from 𝑟 equals three to five of negative eight over 𝑟 as negative 94 over 15.