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Question Number 140749 by SOMEDAVONG last updated on 12/May/21

$$\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}{5}^{\frac{1}{\mathrm{k}}}(1-5^{-\frac{1}{\mathrm{k}(\mathrm{k}+1)}})=?$$

Answered by Dwaipayan Shikari last updated on 12/May/21

$$5^{\frac{1}{k}}(1-5^{-\frac{1}{k}+\frac{1}{k+1}})=5^{\frac{1}{k}}-5^{\frac{1}{k+1}}$$$$\underset{k=1}{\sum ^n}(5^{\frac{1}{k}}-5^{\frac{1}{k+1}})=5-\sqrt{5}+\sqrt{5}-\sqrt[3]{5}+..+5^{\frac{1}{n}}-5^{\frac{1}{n+1}}=5-5^{\frac{1}{n+1}}$$

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