Question Video: Evaluating the Sum of a Finite Series after Expanding It | Nagwa Question Video: Evaluating the Sum of a Finite Series after Expanding It | Nagwa

Question Video: Evaluating the Sum of a Finite Series after Expanding It Mathematics • Second Year of Secondary School

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Expand and then evaluate ∑_(𝑟 = 1)^(4) (19𝑟²).

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Video Transcript

Expand and then evaluate the sum from 𝑟 equals one to 𝑟 equals four of 19𝑟 squared.

Our first step in order to expand is to write the four terms from 𝑟 equals one to 𝑟 equals four. When 𝑟 is equal to one, 19𝑟 squared is equal to 19 multiplied by one squared. When 𝑟 equals two, we have 19 multiplied by two squared. When 𝑟 equals three and 𝑟 equals four, we have 19 multiplied by three squared and 19 multiplied by four squared, respectively. As we need to evaluate the sum, we need to add all four of these terms. We could do this by typing each one individually into the calculator.

However, we might notice that each of them has a common term of 19. This means that we could factor or factorize 19 out of the parentheses. We would then be left with one squared plus two squared plus three squared plus four squared inside the parentheses. One squared is equal to one, two squared is equal to four, three squared is equal to nine, and four squared is equal to 16. Adding these four numbers gives us 30. This means that the sum of 19𝑟 squared between 𝑟 equals one and 𝑟 equals four is 19 multiplied by 30. As 19 multiplied by three is 57, our final answer is 570.

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