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Express the series 54 × 12 + 54 × 24 + 54 × 36 + ... + 54 × 240 in sigma notation.

Express the series 54 multiplied by 12 plus 54 multiplied by 24 plus 54 multiplied by 36, and so on, all the way up to 54 multiplied by 240 in sigma notation.

In this question, we are given the first three terms of our series. Together with the 𝑛th or final term. We have been asked to express this series using sigma notation as shown. We need to find an expression for the general term 𝑢 sub 𝑟 and the number of terms in the series 𝑛. We notice that each of our terms is the product of two numbers. The first number is 54 in each of our terms. The second number corresponds to the multiples of 12. We have 12 times one, 12 times two, 12 times three, and so on. The final term is equal to 12 multiplied by 20. This means that the general term for our series is 54 multiplied by 12𝑟 where 𝑟 is a variable going from one to 20.

54 multiplied by 12 is equal to 648. Therefore, 54 multiplied by 12𝑟 is 648𝑟. Using sigma notation, we can therefore express the series as the sum of 648𝑟 where 𝑟 takes values from one to 20.

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