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📰 Solution: Use partial fractions: $ \frac{1}{k(k+2)} = \frac{1}{2}\left( \frac{1}{k} - \frac{1}{k+2} \right) $. The sum becomes $ \frac{1}{2} \left( \sum_{k=1}^{50} \frac{1}{k} - \sum_{k=1}^{50} \frac{1}{k+2} \right) = \frac{1}{2} \left( \sum_{k=1}^{50} \frac{1}{k} - \sum_{k=3}^{52} \frac{1}{k} \right) $. Telescoping gives $ \frac{1}{2} \left( \frac{1}{1} + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) = \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \cdot \frac{3975}{2652} = \frac{1325}{1768} $. \boxed{\dfrac{1325}{1768}} 📰 Question: Let $ p $ and $ q $ be complex numbers such that $ p + q = 3 + 4i $ and $ p^2 + q^2 = 10 - 24i $. Find $ pq $. 📰 Solution: Use identity $ p^2 + q^2 = (p + q)^2 - 2pq $. Substituting: $ 10 - 24i = (3 + 4i)^2 - 2pq $. Compute $ (3 + 4i)^2 = 9 + 24i - 16 = -7 + 24i $. Then $ 10 - 24i = -7 + 24i - 2pq $. Rearranging: $ 17 - 48i = -2pq \Rightarrow pq = \frac{-17 + 48i}{2} $. \boxed{-\dfrac{17}{2} + 24i} 📰 Stop Strugglingtapmad Delivers Life Changing Results Instantly 1091567 📰 Dont Skip The Details How Much Is Disney Plus Actually Charging 952132 📰 Atlas Fell The Shocking Truth Behind This Mountains Devastating Collapse 1689285 📰 Assassins Creed Shadows Buy 3967281 📰 Graceland Cast 1847673 📰 Film Rush Cast 5697787 📰 Fonendi Secret That Will Transform Your Life Overnight 8124876 📰 Sickness In Spanish 2343652 📰 Unlock The Secret Breakfast That Jack In The Box Hides Behind Every Box 3032563 📰 Gluteal Fold 4532659 📰 Student Loan With Bad Credit 170856 📰 From Page To Power Knight Stevens Shocking Adventures You Need To Watch 6000747 📰 Jumanji 3 Revealed The Dark Secrets That Changed Action Movies Forever 2133842 📰 Sounds Of Freedom 2413026 📰 Kyiv 4234090