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: 1 + 3 + 5 + ... + (2n - 1) = n^2

Accepted Answer

hey everyone here we are asked to write as sub one. Sub two and sub three for the given statement. S sub N. As well as prove that these three statements are true. So to begin this problem we want to find the values for each of these statements using the right hand side of our given state. So we can begin by finding our value for S. Sub one. And to do this we again look at our given statement and use the expression on the right hand side which is N squared plus N. Now we need to replace N with our subscript which in this case is one, so we'll have N be replaced by one. Next we can write our expression so we have one squared plus one. And so we see that S. Sub one is just equal to two. Now we can move on to finding S. Sub to repeating the same process. We replace N with the subscript which is two and we use the expression on the right hand side of the statement. So we have two squared plus two which simplifies to six. Now we can move on to our third and final statement. S sub three repeating this process, we replace N with three and so we have three squared plus three simply fine. We see that S sub three is equal to 12. Next we need to prove that each of these three statements are true. So to do this we now look at the series on the left hand side of our given statement and we check each of the three statements. So beginning with our first statement as sub one, we know that S. Sub one is the first term in the series. So we can look at our given statement and see that S. Sub one. Our first term is two. So now we can see that from the right hand side of the statement we got the value to and from the left hand side. The value is also to for our first statement. Therefore this statement is true. Now we can move on to S sub two. Again we now are looking at the series on the left hand side. So S sub two is the sum of the first two terms. So S sub two is two plus four which simplifies to six and now we can see that both of our values for S sub two are the same. Therefore our second statement is also true. Now we can move on to our last statement as sub three as sub three is going to be the sum of the first three terms. So we we have two plus four plus six which simplifying, we get 12. And now we can see that from the left hand side and the right hand side. S sub three is equal to 12. Therefore this statement is also true and with this we have proved that each of our three statements are true and we are left with answer choice. D 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: 1 + 3 + 5 + ... + (2n - 1) = n^2

Key Concepts

Arithmetic Series

Mathematical Induction

Quadratic Functions

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