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 is a factor of n^3 - n.

Accepted Answer

Hey everyone here we are asked to write as sub one, Sub two and S sub three for the given statement as sub N. And prove that these three statements are true. So we can begin this problem with finding S. Sub one. And now here we need to look at our given statement of S sub N. Where we see that seven is a factor of N. To the seventh power minus N. So here our subscript for S sub one is one, so we will replace N with one. And this gives us as sub one being one to the seventh power minus one and this is just one minus one. So we see that S sub one is equal to zero. Now we can move on to S sub to repeating this process. We look at our given statement and we replace N with now to so as sub two is two to the seventh power minus two simplifying. We see that S sub two is equal to 126. Finally moving on to S sub three, we replace N with this sub subscript which is three. So we have three to the seventh power minus three which simplifies to 2100 And 84. Now we have found our three statements here and we know that seven is indeed a factor of 0, 126 as well as 2,184. Therefore we have proved that these three statements are true and with this 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 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 is a factor of n^3 - n.