A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true.
Sn: 2 is a factor of n2 + 7n
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Could you please show your work? I want to know how to solve.
Just substitute in values for n:
Sn: 2 is a factor of n^2+7n
S1: 2 is a factor or 1^2+7*1
S2: 2 is a factor of 2^2+7*2
S3: 2 is a factor of 3^2+7*3
Now just show that each is true.
To write the statements S1, S2, and S3 and demonstrate their truth, we need to substitute different positive integer values into Sn and verify if 2 is indeed a factor of each expression.
Statement S1: Substitute n = 1 into Sn:
S1: 2 is a factor of (1^2 + 7*1) = 2 is a factor of 8.
Since 8 can be expressed as 2 * 4, S1 is true.
Statement S2: Substitute n = 2 into Sn:
S2: 2 is a factor of (2^2 + 7*2) = 2 is a factor of 18.
Since 18 can be expressed as 2 * 9, S2 is true.
Statement S3: Substitute n = 3 into Sn:
S3: 2 is a factor of (3^2 + 7*3) = 2 is a factor of 36.
Since 36 can be expressed as 2 * 18, S3 is true.
By substituting different positive integer values into Sn and verifying that the resulting expressions are divisible by 2 (i.e., 2 is a factor), we have shown that S1, S2, and S3 are all true.