In Exercises 5–10, a statement Sn about the positive integers is ...

Question

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely.

Sn: 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Accepted Answer

Hey everyone here we are asked to find S. Sub K and S sub K plus one. For the given statement S sub N where n is any positive imager. And we also need to simplify as sub K plus one fully. So our given statement here of S sub N. We have six plus 11 plus 16 plus some other values plus the quantity of five N plus one. And this is equal to the quantity of five N squared plus seven N. All divided by two. And so now we can begin solving this problem by first finding our statement for S sub K. And to do this, all we have to do is substitute in K. For anywhere we have N. In our given statement. So doing this, our S sub K statement becomes six plus plus 16 plus the quantity of five K plus one. And this is equal to the quantity of five K squared plus seven K. All divided by two. And so now there aren't any simplifications to be made to this statement. So we can go ahead and move on to finding our second statement of S sub K plus one. S sub k plus one. So here to find a statement we just repeat the same thing we did for our S sub k statement but instead we substitute in K plus one for N. So doing this, we get the statement of six plus 11 plus plus the quantity of five times the quantity of K plus one plus one. And this is equal to the quantity of five times the quantity of K plus one squared plus seven times the quantity of K plus one. And this is all divided by two. So now we just have to simplify our statement here and doing this, we see that S sub K plus one is the statement of six plus 11 plus 16 plus the quantity of five K plus six. And this is equal to the quantity of five K squared plus 17 K plus 12, all divided by two. So with this we have found our statement for S sub K and we have also found our simplified statement for S sub K plus one and therefore we are left here with answer choice. See thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely. Sn: 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Key Concepts

Arithmetic Series

Mathematical Induction

Algebraic Manipulation

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