### Video Transcript

Express the series 32 times 33 plus 33 times 34 plus 34 times 35 and so on in sigma notation.

Let’s begin by reminding ourselves what it means to express a series in sigma notation. When we represent something using sigma notation, we’re talking about finding its sum. We define an upper and lower limit and some function for which we’re finding the sum of. So to be able to express our series in sigma notation, let’s begin by seeing if we can identify what the function is.

Let’s begin also by assuming that the lower limit of our notation is going to be equal to one. This is an arbitrary choice, but it’s certainly the most conventional value to pick in the absence of any other information. With this in mind, the term 32 times 33 is going to be 𝑓 of one, the term 33 times 34 is going to be 𝑓 of two, and so on. Let’s see if we can identify the 𝑛th term of this series.

Well, in fact, if we look closely, we see we can write 𝑓 of one as one plus 31 times one plus 32. We can write 𝑓 of two as two plus 31 times two plus 32 and so on. So 𝑓 of 𝑛 must be 𝑛 plus 31 times 𝑛 plus 32. Now, as we said earlier, it’s convention to use 𝑓 of 𝑟. And this series appears to continue forever. So the lower and upper limit of our summation are going to be one and ∞, respectively. So the series is the sum from 𝑟 equals one to ∞ of 𝑓 of 𝑟. And since 𝑓 of 𝑛 is 𝑛 plus 31 times 𝑛 plus 32, 𝑓 of 𝑟 must be 𝑟 plus 31 times 𝑟 plus 32. And so we can express this series in sigma notation as the sum from 𝑟 equals one to ∞ of 𝑟 plus 31 times 𝑟 plus 32.