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: 4 + 8 + 12 + ... + 4n = 2n(n + 1)

Accepted Answer

Hey everyone here we are asked to write S sub K and S sub K plus one. For the given statement S sub N where n is any positive integer. So we can begin this problem by writing out S sub K. Now here the key is is we need to look at our given statement here and inside instead of the variable N. We're going to have K. So N is going to be replaced by K. So with this our S sub k statement is five plus 10 plus 15 plus some other value plus five times K instead of N is equal to five halves times K times the quantity of K plus one. And now we can move on to the second statement here. We have S sub K plus one. And so we just repeat what we did earlier. We have to look at our given statement and instead of N we're going to replace it with K plus one. So doing this we have five plus 10 plus 15 plus some other value plus five times the quantity of now K plus one is equal to five halves times the quantity of K plus one times the quantity of K plus one plus one. And so now we are asked to simplify this statement fully. So we'll rewrite this in its simplified form. We have again S sub K plus one and our statement here is just five plus 10 plus 15 plus some other value plus five times the quantity of K plus one is equal to five halves times the quantity of K plus one. And now just simplifying this last set of parentheses, we have K plus two and with this we have our two statements and we are left with answer choice B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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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: 4 + 8 + 12 + ... + 4n = 2n(n + 1)