Exercise 2: A Triangular Loop at n=10 — Possible Solution
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GIVEN
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A triangular loop identical in structure to this chapter's own
example (inner loop running from 0 to i-1 for each outer value i),
with n = 10.

STEP 1: DIRECT SUMMATION
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Total operations = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

  = 45

STEP 2: THE CLOSED-FORM FORMULA
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Per this chapter's own formula (reusing Discrete Mathematics
Fundamentals Chapter 8's own summation result):

  n(n-1)/2 = 10(9)/2 = 90/2 = 45

STEP 3: CONFIRMING THE MATCH
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Direct summation gives 45. The closed-form formula also gives 45 -
an exact match, confirming the formula correctly predicts the total
operation count without needing to add up every term by hand.

STEP 4: THE BIG-O CLASS
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Per this chapter's own dominant-term simplification of n(n-1)/2
(which expands to (n^2-n)/2), the dominant term is n^2, so the
overall complexity is:

  O(n^2)

WHY THIS WORKS AS AN ANSWER
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Both the direct summation and the closed-form formula are computed
independently and explicitly shown to agree at n=10, confirming this
chapter's own formula is correct rather than merely assumed, and the
Big-O class is derived using the same dominant-term reasoning applied
throughout this chapter.