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: 2 is a factor of n^2 - n + 2.

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 we have to is a factor of n squared plus three N. So to begin we can start with S sub K and well we have to look at our given statement which has a variable end here. And all we have to do is replace the end with K. So we'll have N is equal to K. And all we have to do is substitute in K. So we'll have S sub K where we have to is a factor of and again, instead of N, we have K squared plus three K. And now we can simply move on to the next term next statement here we have S sub K plus one. And just like before we look at our given statement and instead of n we'll have K plus one. So again we'll have that too is a factor of and instead of n we have the quantity of K plus one squared plus three times the quantity of K plus one. But in our problem statement here we're asked to simplify as sub k plus one fully. So we our statement here is S sub K plus one where two is a factor of and now simplifying this by squaring this expression here and factoring in the three, we are left with K squared plus five K plus four. And with this we have found our two statements which leaves us with answer choice D. 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: 2 is a factor of n^2 - n + 2.

Key Concepts

Factors

Mathematical Induction

Quadratic Expressions

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