In Exercises 1–4, a statement S_n about the positive integers is ...

Question

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true.

Sn: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

Accepted Answer

Hey everyone here we are asked to write as sub one. Sub two and sub three as well as prove that these three statements are true for the given statement as sub N where n is any positive imager. So we can begin with finding the value for S sub one. And to do this we first need to look at the right hand side of our given statement here which is end times the quantity of N plus three divided by two. And we need to take this expression and replace the end variables with our subscript which in this case is one so and will be replaced by one. And now we see that S sub one is equal to one times the quantity of one plus three divided by two. And simplifying we see that S sub one is equal to two. Now we can move on to finding the value for S sub two. And again we use the subscript to replace the end variable so N will be replaced by two and we need to use the expression on the right hand side of our given statement. So we have two times the quantity of two plus three divided by two simplifying. We see that as sub two is equal to five and now we can move on to our third statement S sub three. Again we replace N with the subscript which in this case is three. So we have three times the quantity of three plus two divided by two which gives us S. Sub three is equal to nine. And now we have found the values for each of our statements here but our next step is to prove that these statements are true. So to do this we need to look at our left hand side of the given statement which is a series and compare our statements. So we can begin with S sub one again. And in this case as sub one is our first term. So looking at the first term in the series, we see that it is indeed too. So both of our terms are the same. So we know that our first statement is true. Now we can move on to S sub two. Well S sub two is the song of the firm first two terms. So we have S sub two is two plus three and simplifying S sub two is equal to five again both of the values are the same from what we found now to what we found before from the right hand side. So this second statement is also true. Now we can move on to our third statement as sub three as sub three is going to be the sum of the first three terms. So we have two plus three plus four and simplifying this, we see that as sub three is equal to nine. Both of the valley are the same from the left hand side and the right hand side for the S sub three statement. So this statement is also true and with this we have verified each of our three statements and found each of their values. And with this we are left with answer choice. C. Thanks for watching. I hope you found this video helpful, and I'll see you again next time.

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true. Sn: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

Key Concepts

Mathematical Induction

Summation of Series

Algebraic Manipulation

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