In Exercises 29–42, find each indicated sum. 5Σi=1 i!/(i−1)!

Question

Accepted Answer

Hello today we're going to be evaluating the indicated summation. So what it means to evaluate a summation is we're going to plug in all the possible values for our index directly into the pattern of the summation. So let's take a look at the values of the index Here. The starting value for the index is going to be X is equal to three. So the first value of X is going to be three and the final value of X can be found at the top of the sigma which in this case is going to be seven. So we're gonna be plugging in the values of X starting with three and going all the way to seven. And where are we going to be plugging in the values of X? Where we're gonna be taking these values of x and plugging them into the given pattern for the summation notation and the given pattern is x factorial over X - factorial. So in order to find the sum, we need to first find each term of the values of the index. And we're going to start by finding the term when x is equal to three. When X is equal to three. Plugging this into the pattern is going to give us X factorial which is three factorial over x minus three which is three minus three factorial. Simplifying This. The numerator gives us three factorial over 3 -3 which is zero factorial. And recall that the value of zero factorial is always going to be one. So the numerator denominator is going to be one and we can expand the numerator to be three times two times one and that is going to give us 6/1 which is just six. So this is the first term of our some and we want to go ahead and repeat this process for the other values of the index. When X is equal to four We get four factorial over 4 -3 factorial simplifying this, we get four factorial over four minus three which is just one factorial and one factorial just simplifies to one, expanding the numerator is going to give us four times three times two times one. All of that over one. And that's going to give us the value of 24. Next When X is equal to five We get five factorial over 5 -3 factorial And simplifying this is going to give us five factorial over 5 - which is two factorial. Now we can we can expand the numerator to give us five times four times three times two factorial and that's going to be over two factorial. Since we have a two factorial in both the numerator and denominator we can cancel those out to leave us with five times four times three which is going to give us the value of 60. Next we're gonna plug in X is equal to six and that's going to give us six factorial over six minus three factorial simplifying is going to give us six factorial over six minus three which is three factorial And if we expand the numerator we get six times five Times four times 3 factorial. All over three factorial. Since we have the three factorial, both numerator and denominator. Those cancel out to leave us with six times five times four which is going to give us the value of 120. And finally when X is equal to seven We get seven factorial over 7 -3 factorial. And simplifying. This is going to give us seven factorial over 7 -3 which is four factorial expanding the numerator is going to give us seven times six times five times four factorial over four factorial. And we can cancel out the four factorial in both the numerator and denominator to leave us with six times 567 times six times five. Which is going to give us 210. So now what we want to go ahead and do is take all the terms that we just calculated and put them together as a sum and that's going to give us six plus 24 plus plus 100 and 20 plus 210. And adding all these values together is going to give us the total sum of 420 which is going to be the answer to this problem. And that means that the solution to the problem is going to be C. So I hope this video helps you in understanding how to evaluate a summation, and I'll go ahead and see you all in the next video.

In Exercises 29–42, find each indicated sum. 5Σi=1 i!/(i−1)!

Key Concepts

Factorial Notation

Summation Notation

Simplifying Factorials

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