Express the series 26 plus 27 plus 28 plus 29 plus 30 all the way up to 84 in ∑ notation.
This Greek letter ∑ tells us to find the sum of something. In particular, suppose we have a function or sequence that’s denoted 𝑓 of 𝑟 where 𝑟 is an integer. Then, for two numbers 𝑎 and 𝑏, where 𝑎 is smaller than 𝑏, we can sum this using the given notation, the sum from 𝑟 equals 𝑎 to 𝑏 of 𝑓 of 𝑟. This then tells us that we evaluate the expression 𝑓 of 𝑟 for all values 𝑎 to 𝑏 inclusive and find their sum. So we get 𝑓 of 𝑎 plus 𝑓 of 𝑎 plus one all the way up to 𝑓 of 𝑏. And so, it follows that if we’re given the function and the values for 𝑎 and 𝑏, we can evaluate the sum by substituting those values into the function, all values in between, and finding the sum.
But how do we reverse that when we’re given a series, here the series from 26 up in ones all the way up to 84? Well, we know that these are terms in a sequence. We don’t know what the first term is. It could be 26 but it might not. So let’s consider what could be happening in this sequence. We can see that the common difference between terms is one. So we have a linear or arithmetic sequence of some sort. Since the common difference is one, we ask ourselves, “Well, could the 𝑛th term be 𝑛?” Well, yes, it could. And that would assume that the first term in the sequence is one. This then means that the term 26 is the 26th term in the sequence. The term 27 is the 27th term and so on.
So let’s define the function that describes our sequence as 𝑓 of 𝑟 equals 𝑟. Now, we’re using 𝑟 because that’s generally the letter we use when dealing with ∑ notation. Since we want to find the sum of the series beginning with the 26th term and ending with the 84th, we can say that the series given is the sum from 𝑟 equals 26 to 84 of 𝑟. And then we can check this gives us the relevant series by evaluating 𝑟 at 26, 27, all the way up to 84 and finding their sum. This matches the series given in our question, so we’re done. The series is the sum from 𝑟 equals 26 to 84 of 𝑟.