A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.

Sn:   2 is a factor of n2 + 7n

Wow! I haven't seen an identity crisis like this in quite a while!!

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Dr. Lexi
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Plus ... both Steve and Reiny already answered you on this:
http://www.jiskha.com/display.cgi?id=1457118310

Sn: 2 is a factor of n2 + 7n

n is either even or odd.

If n is even, n=2k
n^2 + 7n
= (2k)^2 + 7(2k)
= 4k^2 + 14k
= 2(2k^2 + 7k)
2 is a factor

If n is odd, n=2k+1
n^2 + 7n
= (2k+1)^2 + 7(2k+1)
= 4k^2+4k+1 + 14k+7
= 4k^2+18k+8
= 2(2k^2+9k+4)
2 is a factor

So, Sn is true

To show that a statement is true, we need to prove it by providing a logical and valid mathematical argument. In this case, we have the statement Sn: 2 is a factor of n^2 + 7n.

Statement S1: n is an even number.
To prove S1 is true, we need to show that if n is even, then 2 is a factor of n^2 + 7n.

For any even number n, we can write it as n = 2k, where k is an integer.

Substituting n = 2k into the expression n^2 + 7n, we get:
(2k)^2 + 7(2k) = 4k^2 + 14k = 2(2k^2 + 7k)

Here, we can see that 2 is a common factor of the expression 2k^2 + 7k because we can factor out the 2.

Therefore, if n is an even number, Sn is true.

Statement S2: n is an odd number.
To prove S2 is true, we need to show that if n is odd, then 2 is a factor of n^2 + 7n.

For any odd number n, we can write it as n = 2k + 1, where k is an integer.

Substituting n = 2k + 1 into the expression n^2 + 7n, we get:
(2k + 1)^2 + 7(2k + 1) = 4k^2 + 4k + 1 + 14k + 7 = 4k^2 + 18k + 8 = 2(2k^2 + 9k + 4)

Here, we can see that 2 is a common factor of the expression 2k^2 + 9k + 4 because we can factor out the 2.

Therefore, if n is an odd number, Sn is true.

Statement S3: n can be any positive integer.
To prove S3 is true, we need to show that for any positive integer n, 2 is a factor of n^2 + 7n.

We have already proved S1 and S2 for even and odd numbers, respectively. Since every positive integer is either even or odd, S3 is automatically true based on the proofs of S1 and S2.

Therefore, Sn: 2 is a factor of n^2 + 7n is true for all positive integers.