Express the series eight plus 98 plus 998 plus 9998, and so on in ∑ notation to 𝑛 terms.
Remember, we can describe the sum of a function or sequence using ∑ notation. Let’s begin by supposing the sequence is denoted 𝑓 of 𝑟, where 𝑟 here is an integer. For two integers 𝑎 and 𝑏 where 𝑏 is greater than or equal to 𝑎, we can sum the values of this function using the notation shown. This is read as the sum from 𝑟 equals 𝑎 to 𝑏 of 𝑓 of 𝑟. And so, we have a series, which is the sum of the terms 8, 98, 998, and so on. We need to find a way to describe each term in this series as some function in 𝑟.
Without any extra information, it makes sense to choose the first term to be 𝑓 of one. Then, 𝑓 of two in this sequence is 98, 𝑓 of three is 998, and so on. Now, if we look carefully, we can see that each term is related to a power of 10. In particular, the first term 𝑓 of one is 10 to the first power minus two. That does indeed give us eight. The second term 𝑓 of two is 10 squared minus two. That’s 100 minus two, which is indeed 98. In a similar way, we can write the third term 998 as 10 cubed minus two and the fourth term as 10 to the fourth power minus two.
Since we’re looking to express the series up to 𝑛 terms, we want to find the 𝑛th term. By following the pattern that we’ve written, we can say that 𝑓 of 𝑛 must be 10 to the 𝑛th power minus two. And so, our function itself must be 10 to the 𝑟th power minus two.
Since we defined the first term to be 𝑓 of one and the last term to be 𝑓 of 𝑛, our ∑ notation is as follows. It’s the sum from 𝑟 equals one to 𝑛 of 10 to the 𝑟th power minus two. Now, it’s worth noting that it’s sensible to represent this by enclosing the expression in parentheses so that we know we’re summing the negative two as well.