### Video Transcript

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.