### 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.