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 n^2 + 7n
S1:
2 | 1^2 + 7*1
2 | 1+7
2 | 8
true
do likewise for n=2,3
To show that each of the statements S1, S2, and S3 is true, we need to substitute different positive integer values in place of n in the given statement Sn and check if 2 is a factor of the resulting expression.
Statement S1: Substitute n = 1
S1: 2 is a factor of (1^2 + 7*1)
Simplifying, we get:
S1: 2 is a factor of (1 + 7)
S1: 2 is a factor of 8
To determine if 2 is a factor of 8, we check if 8 is divisible by 2. Since 8 divided by 2 equals 4 with no remainder, 2 is indeed a factor of 8. Therefore, S1 is true.
Statement S2: Substitute n = 2
S2: 2 is a factor of (2^2 + 7*2)
Simplifying, we get:
S2: 2 is a factor of (4 + 14)
S2: 2 is a factor of 18
To determine if 2 is a factor of 18, we check if 18 is divisible by 2. Since 18 divided by 2 equals 9 with no remainder, 2 is indeed a factor of 18. Therefore, S2 is true.
Statement S3: Substitute n = 3
S3: 2 is a factor of (3^2 + 7*3)
Simplifying, we get:
S3: 2 is a factor of (9 + 21)
S3: 2 is a factor of 30
To determine if 2 is a factor of 30, we check if 30 is divisible by 2. Since 30 divided by 2 equals 15 with no remainder, 2 is indeed a factor of 30. Therefore, S3 is true.
By substituting different positive integer values for n in the original statement Sn and verifying that 2 is a factor of the resulting expressions, we have shown that S1, S2, and S3 are all true.