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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} 📰 Most Efficient Desk 5848776 📰 You Wont Guess Which Foods Afternoon Hunger Returns To Fastheres What Science Says 5477295 📰 Aks Le Shows Bombshell Twist What Viewers Are Saying Is Unreal 9957106 📰 Hyatt Herald Square 2836812 📰 Stroke Luck Discover The 5 Hidden Resources That Boost Your Income Fast 9365022 📰 Trump Mobiles New Smartphone Release Has Been Significantly Delayed 9630884 📰 Touted As Next Big Thing Yellow Top Clothing Is Stirring Up Every Shop 7672987 📰 Wells Fargo Flanders Nj 3450222 📰 00002 1050 00002 500 410 1C 2219539 📰 Ground News Review The Hidden Agenda You Need To Know Before Todays Headline 9501297 📰 Nsa Warning Update 5185505 📰 Lilys Garden Revealed The Hidden Oasis Youll Never Believe Exists 4084675 📰 Gorillaz Fortnite The Most Surprising Collaboration That Will Make You Go Wow 9152763 📰 Best Western Corpus Christi 5554717 📰 The Shocking Truth Behind Be Soccer Why Every Fans Craving More 2264099