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PrecalculusSuperchallengeWhy the Formulas Work

Why the Formulas Work · Superchallenge problem

Evaluate 11⋅2+12⋅3+⋯+153⋅54\dfrac{1}{1\cdot 2} + \dfrac{1}{2\cdot 3} + \cdots + \dfrac{1}{53\cdot 54}.

Express your answer as a common fraction.