Final answer:
The student's question contains a misconception: the formula given, 1/2 n(n - 1), is for the number of distinct pairs in a set, not the sum of whole numbers. The correct formula for the sum of the first n positive whole numbers is 1/2 n(n + 1), and substituting n=2,3,4,5 into this correct formula yields the correct sums, which are 3, 6, 10, and 15 respectively.
Step-by-step explanation:
A student asked to verify that the formula 1/2n(n−1) produces the sum of the first n positive whole numbers. To clarify, this is actually a misunderstanding. The correct formula that represents the sum of the first n positive whole numbers is 1/2 n(n + 1), not 1/2 n(n - 1). The latter formula actually calculates the number of distinct pairs that can be formed from n items, which is also known as the number of edges in a complete graph of n vertices, related to combinatorial mathematics. However, substituting n=2,3,4,5 into 1/2 n(n - 1) will not yield the sums of whole numbers but rather 1, 3, 6, 10 which coincidentally are the same as the triangular numbers sequence.
To correct the misunderstanding, let's use the appropriate formula for calculating the sum of the first n positive whole numbers: 1/2 n(n + 1). Here's how it works for n=2,3,4,5:
- n=2: 1/2 * 2(2 + 1) = 1/2 * 2(3) = 3 − the sum of the first two whole numbers (1+2).
- n=3: 1/2 * 3(3 + 1) = 1/2 * 3(4) = 6 − the sum of the first three whole numbers (1+2+3).
- n=4: 1/2 * 4(4 + 1) = 1/2 * 4(5) = 10 − the sum of the first four whole numbers (1+2+3+4).
- n=5: 1/2 * 5(5 + 1) = 1/2 * 5(6) = 15 − the sum of the first five whole numbers (1+2+3+4+5).
Clearly, the correct formula 1/2 n(n + 1) does indeed produce the sum of the first n positive whole numbers.