2002 AMC 12P Problems/Problem 11
Contents
[hide]Problem
Let be the th triangular number. Find
Solution 1
We may write as and do a partial fraction decomposition. Assume .
Multiplying both sides by gives .
Equating coefficients gives and , so . Therefore, .
Now
Note: For the sake of completeness, I put the full derivation of the partial fraction decomposition of here. However, on the contest, the decomposition step would be much faster since it is so well-known.
Solution 2 (Cheese)
As with all telescoping problems, there is a solution that involves induction. In competition, it is sufficient to conjecture the formula but not prove it. For sake of completeness and practice, we will prove the formula for
With some guess and check:
From and we can conjecture A quick check shows that for gives which means our inductive hypothesis is most likely correct. Thus, our answer is
We will prove this with induction for all as promised.
Base case:
Since the base case is proven.
Induction step:
Assume:
We want to prove:
We are given Add on both sides and simplify.
Given the inductive hypothesis, we have proven the induction step. Thus, we have completed our proof for induction.
See also
2002 AMC 12P (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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