Question Video: Evaluating Series Algebraically Mathematics

Find ∑_(𝑟 = 8) ^(12) 9(𝑟 − 37) using the properties of summation and given ∑_(𝑟 = 1) ^(𝑛) (𝑟) = (𝑛(𝑛 + 1))/2.

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Video Transcript

Find the sum of nine multiplied by 𝑟 minus 37 from 𝑟 equals eight to 12 using the properties of summation and given the sum of 𝑟 from 𝑟 equals one to 𝑛 is 𝑛 multiplied by 𝑛 plus one divided by two.

We will begin by distributing our parentheses so that the linear expression is equal to nine 𝑟 minus 333. We need to calculate the sum of this from 𝑟 equals eight to 𝑟 equals 12 using the properties of summation.

When our starting index is greater than one, we can use the difference of two series property. As our value of 𝑚 is equal to eight and 𝑛 is equal to 12, we can rewrite the expression as shown. Next, we can use the linear property of summation. Recalling that when subtracting a negative number we get a positive answer, we are left with nine multiplied by the sum of 𝑟 from 𝑟 equals one to 12 minus 333 multiplied by the sum of one from one to 12 minus nine multiplied by the sum of 𝑟 from 𝑟 equals one to seven plus 333 multiplied by the sum of one from 𝑟 equals one to seven.

We are given an expression for the sum of 𝑟 from 𝑟 equals one to 𝑛. And in this question, 𝑛 will be equal to 12 and seven, respectively. This means that our first term is equal to nine multiplied by 12 multiplied by 13 divided by two. And the third term is equal to nine multiplied by seven multiplied by eight divided by two. These simplify to nine multiplied by 78 and nine multiplied by 28.

We recall that the sum of one from 𝑟 equals one to 𝑛 is equal to 𝑛. Our expression therefore becomes nine multiplied by 78 minus 333 multiplied by 12 minus nine multiplied by 28 plus 333 multiplied by seven. This is equal to negative 1,215. This is the sum of nine multiplied by 𝑟 minus 37 from 𝑟 equals eight to 12.

An alternative method here would be to simply substitute the integers from eight to 12 into our expression and then find the sum of these values.