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: 2 is a factor of 1^2+7*1
and so on.
Clearly, since
if n is odd, n^2 and 7n are both odd, so their sum is even
if n is even. n^2 and 7n are both even, so their sum is even.
Or,
if n is odd
(2k+1)^2+7(2k+1) = 4k^2+4k+1+14k+7 = 2(2k^2+9k+4)
and similarly for even n=2k
To prove that the statement Sn is true, we need to demonstrate that the given expression, n^2 + 7n, is divisible by 2 for three different positive integers.
Statement S1: Substitute n = 1 into the expression n^2 + 7n:
S1: 1^2 + 7(1) = 1 + 7 = 8
Since 8 is divisible by 2 (8 ÷ 2 = 4), S1 is true.
Statement S2: Substitute n = 2 into the expression n^2 + 7n:
S2: 2^2 + 7(2) = 4 + 14 = 18
Since 18 is divisible by 2 (18 ÷ 2 = 9), S2 is true.
Statement S3: Substitute n = 3 into the expression n^2 + 7n:
S3: 3^2 + 7(3) = 9 + 21 = 30
Since 30 is divisible by 2 (30 ÷ 2 = 15), S3 is true.
By demonstrating that S1, S2, and S3 are all true, we have proven that the statement Sn: "2 is a factor of n^2 + 7n" is true for these three cases.