Video Transcript
Evaluate the sum from 𝑟 equals one to nine of five.
We recall first that this notation means that we are summing a term, in this case five, from 𝑟 equals one to 𝑟 equals nine. Now, usually the term we’re summing, the summand, would be a function of 𝑟, but in this case, it is just a constant. We can therefore recall a key property of summation. The sum from 𝑟 equals one to 𝑛 of 𝛼, where 𝛼 is a constant, is equal to 𝛼 multiplied by 𝑛.
In this question, the value of 𝛼 we’re summing is five, and the value of 𝑛, the finishing index for 𝑟, is nine. So, applying the standard result, we have that the sum from 𝑟 equals one to nine of five is equal to five multiplied by nine, which is 45.
We can check this by direct calculation if we write out the series in longhand. The sum from 𝑟 equals one to nine of five is simply five plus five plus five plus five plus five plus five plus five plus five plus five. We’re adding nine lots of five. Once again, that’s nine times five or five times nine, which is equal to 45.
Using direct calculation and using one of the key properties of summation, we found that the sum from 𝑟 equals one to nine of five is 45.
