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Find ∑_(𝑟 = 1) ^(𝑛) (𝑟 − 8) given ∑_(𝑟 = 1) ^(𝑛) (𝑟) = (𝑛(𝑛 + 1))/2.

Find the sum of 𝑟 minus eight from 𝑟 equals one to 𝑛 given the sum of 𝑟 from 𝑟 equals one to 𝑛 is equal to 𝑛 multiplied by 𝑛 plus one divided by two.

We begin this question by recalling the linear property of summation. This means that we can rewrite the summation given as the sum of 𝑟 from 𝑟 equals one to 𝑛 minus eight multiplied by the sum of one from 𝑟 equals one to 𝑛. We are given in the question that the first term on the right-hand side is equal to 𝑛 multiplied by 𝑛 plus one divided by two. Recalling that the sum of one from 𝑟 equals one to 𝑛 is equal to 𝑛, the second term on the right-hand side is equal to eight 𝑛. We have 𝑛 multiplied by 𝑛 plus one over two minus eight 𝑛.

Distributing the parentheses and creating a common denominator, we have 𝑛 squared plus 𝑛 minus 16𝑛 all over two. This in turn simplifies to 𝑛 squared minus 15𝑛 over two. This is the sum of 𝑟 minus eight from 𝑟 equals one to 𝑛.

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