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Question Number 197194 by sonukgindia last updated on 10/Sep/23

Answered by witcher3 last updated on 10/Sep/23

$${\mathrm{k}}^{1-\mathrm{n}}+{(\mathrm{k}+1)}^{1-\mathrm{n}}=\frac{{(1+\mathrm{k})}^{\mathrm{n}-1}+{\mathrm{k}}^{\mathrm{n}-1}}{{\left(\mathrm{k}(1+\mathrm{k})\right)}^{\mathrm{n}-1}}$$$$\iff \underset{\mathrm{k}=1}{\sum ^{50}}\sqrt[\mathrm{n}-1]{\frac{{\mathrm{k}}^{\mathrm{n}-1}+{(1+\mathrm{k})}^{\mathrm{n}-1}}{{\left(\mathrm{k}(\mathrm{k}+1)\right)}^{\mathrm{n}-1}({\mathrm{k}}^{\mathrm{n}-1}+{(1+\mathrm{k})}^{\mathrm{n}-1})}}$$$$=\underset{\mathrm{k}=1}{\sum ^{50}}\frac{1}{\mathrm{k}(\mathrm{k}+1)}=\underset{\mathrm{k}=1}{\sum ^{50}}\frac{1}{\mathrm{k}}-\frac{1}{\mathrm{k}+1}=1-\frac{1}{51}=\frac{50}{51}$$

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