Question Video: Expressing a Given Series in Sigma Notation | Nagwa Question Video: Expressing a Given Series in Sigma Notation | Nagwa

# Question Video: Expressing a Given Series in Sigma Notation Mathematics • Second Year of Secondary School

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Express the series 8 + 98 + 998 + 9,998 + β¦ in sigma notation to π terms.

02:25

### Video Transcript

Express the series eight plus 98 plus 998 plus 9998, and so on in β notation to π terms.

Remember, we can describe the sum of a function or sequence using β notation. Letβs begin by supposing the sequence is denoted π of π, where π here is an integer. For two integers π and π where π is greater than or equal to π, we can sum the values of this function using the notation shown. This is read as the sum from π equals π to π of π of π. 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 π.

Without any extra information, it makes sense to choose the first term to be π of one. Then, π of two in this sequence is 98, π 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 π of one is 10 to the first power minus two. That does indeed give us eight. The second term π of two is 10 squared minus two. Thatβ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 weβre looking to express the series up to π terms, we want to find the πth term. By following the pattern that weβve written, we can say that π of π must be 10 to the πth power minus two. And so, our function itself must be 10 to the πth power minus two.

Since we defined the first term to be π of one and the last term to be π of π, our β notation is as follows. Itβs the sum from π equals one to π of 10 to the πth power minus two. Now, itβs worth noting that itβs sensible to represent this by enclosing the expression in parentheses so that we know weβre summing the negative two as well.

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