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
S1: For n=1, substitute into Sn: 2 is a factor of (1^2) + 7(1) = 1 + 7 = 8. Since 2 is a factor of 8, S1 is true.
S2: Assume Sn is true for some positive integer k: 2 is a factor of k^2 + 7k.
Now, consider k+1:
(k+1)^2 + 7(k+1) = k^2 + 2k + 1 + 7k + 7 = k^2 + 7k + (2k + 8).
Since 2 is a factor of k^2 + 7k (by assuming Sn is true for k), we can write k^2 + 7k as 2m, where m is an integer.
Substitute:
k^2 + 7k + (2k + 8) = 2m + (2k + 8) = 2(m + k + 4).
Since (m + k + 4) is an integer, we have expressed (k+1)^2 + 7(k+1) as a multiple of 2.
Therefore, assuming Sn is true for k and substituting k+1, we can conclude that S2 is true.
S3: By using mathematical induction, we have shown that S1 is true and assuming Sn is true for k implies S2 is true. Therefore, Sn is true for all positive integers, including S3.
To prove that statement Sn is true, we need to find values of n for which the given equation holds true.
Statement S1: 2 is a factor of n^2 + 7n when n = 2.
Let's substitute n = 2 into the equation:
2^2 + 7(2) = 4 + 14 = 18
Since 2 is a factor of 18 (18 ÷ 2 = 9), statement S1 is true.
Statement S2: 2 is a factor of n^2 + 7n when n = 4.
Substituting n = 4 into the equation:
4^2 + 7(4) = 16 + 28 = 44
Since 2 is a factor of 44 (44 ÷ 2 = 22), statement S2 is true.
Statement S3: 2 is a factor of n^2 + 7n when n = 6.
Substituting n = 6 into the equation:
6^2 + 7(6) = 36 + 42 = 78
Since 2 is a factor of 78 (78 ÷ 2 = 39), statement S3 is true.
Therefore, we have shown that each of the statements S1, S2, and S3 is true by substituting values of n that satisfy the given equation and evaluating the results.