Maths Olympiad Prep

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HMMT February

Algebra Difficulty 5.0 AIME, harder Find the answer

Determine the value of the sum 312⋅22+522⋅32+732⋅42+⋯+29142⋅152\frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2} \cdot 3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\cdots+\frac{29}{14^{2} \cdot 15^{2}}

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The sum telescopes as (112−122)+(122−132)+⋯+(1142−1152)=112−1152=224225\left(\frac{1}{1^{2}}-\frac{1}{2^{2}}\right)+\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)+\cdots+\left(\frac{1}{14^{2}}-\frac{1}{15^{2}}\right)=\frac{1}{1^{2}}-\frac{1}{15^{2}}=\frac{224}{225}

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