MATHCOUNTS 2026 State Sprint Round (Problem 30)

ProbabilityVery Hard

Ford has an unfair random number generator that generates positive integers. When the generator is run, each positive integer

n

comes up with probability

\frac{1}{n(n+1)}

. For example, the probability of generating an

8

\frac{1}{72}

. Ford calls a positive integer “interdimensional” if it is one less than a triangular number. (The first few interdimensional numbers are

2,5,9,14,\ldots

.) What is the probability that Ford’s random number generator shows an interdimensional number when run? Express your answer as a common fraction.