### Instruction: A computer program calculated the sum of the first 100 counting numbers as 5,050. Using inductive logic, how can you prove that this is the correct sum?### Response: To prove this using inductive logic, we will use the principle of mathematical induction. The sum of the first n counting numbers can be represented by the formula:

S(n) = n * (n + 1) / 2

We will prove this formula is correct for n = 1 (base case) and then show that if it is true for n = k, it is also true for n = k + 1 (inductive step).

Base case (n = 1):
S(1) = 1 * (1 + 1) / 2 = 1 * 2 / 2 = 1

Since the sum of the first counting number is indeed 1, the base case is true.

Inductive step:
Assume the formula is true for n = k, i.e., S(k) = k * (k + 1) / 2. We need to show that the formula is also true for n = k + 1.

S(k + 1) = S(k) + (k + 1)

Substitute the formula for S(k):
S(k + 1) = k * (k + 1) / 2 + (k + 1)

Factor out (k + 1):
S(k + 1) = (k + 1) * (k / 2 + 1)

Now, simplify the expression inside the parentheses:
S(k + 1) = (k + 1) * ((k + 2) / 2)

This is the same as the formula for S(n) with n = k + 1:
S(k + 1) = (k + 1) * ((k + 1) + 1) / 2

Thus, the formula holds for n = k + 1 if it holds for n = k.

Now, we can use the formula to calculate the sum of the first 100 counting numbers:

S(100) = 100 * (100 + 1) / 2 = 100 * 101 / 2 = 5050

So, the sum of the first 100 counting numbers is indeed 5,050, and the computer program's calculation is correct.