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

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

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Express the series 26 + 39 + 52 + ... + (13𝑛 + 13) in sigma notation.

02:31

Video Transcript

Express the series 26 plus 39 plus 52 up to 13𝑛 plus 13 in sigma notation.

Remember, sigma notation, which is so called because it uses the Greek letter βˆ‘, is a concise way of representing a sum. Specifically, it’s used to describe the sum of some function with an upper and lower limit of 𝑏 and π‘Ž, respectively. To be able to express the series given in this question in sigma notation then, we need to be able to define the function and the upper and lower limits.

Now, in fact, we do appear to be given the 𝑛th term of this series. The final term is 13𝑛 plus 13. We might therefore deduce that the function itself is given by 𝑓 of π‘Ÿ, which is 13π‘Ÿ plus 13. But what would our lower and upper limits be? Well, the final term, we said, is the 𝑛th term. So we can assume that the upper limit for our summation must be 𝑛. But what is our lower limit?

Well, the first term in our series is 26. So our job is to work out which value of π‘Ÿ gives us 26. Let’s set the function 13π‘Ÿ plus 13 equal to 26 then and then solve for π‘Ÿ. To begin, we’ll subtract 13 from both sides. This gives us 13π‘Ÿ equals 13. Next to solve for π‘Ÿ, we’ll divide both sides of the equation by 13. And when we do, we get π‘Ÿ equals one. So we can assume that π‘Ž is one and 𝑏 is 𝑛. So it seems we can express the series given in this question as the sum from π‘Ÿ equals one to 𝑛 of 13π‘Ÿ plus 13.

It might be sensible, though, to double-check that the other terms in our series match this summation. We’ll check by substituting π‘Ÿ equals one, π‘Ÿ equals two, and π‘Ÿ equals three into the expression 13π‘Ÿ plus 13. When we do, we find that the first three terms of the series are 26 plus 39 plus 52. Now this matches the series in our question, so we can deduce that we’ve calculated the correct summation. The series 26 plus 39 plus 52 all the way up to 13𝑛 plus 13 can therefore be written as the sum from π‘Ÿ equals one to 𝑛 of 13π‘Ÿ plus 13.

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