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Author: Tinku Tara

Question-191392

Question Number 191392 by Mingma last updated on 23/Apr/23 Answered by Frix last updated on 23/Apr/23 $$\frac{\sqrt{\mathrm{ln}\:{n}}}{\:\sqrt{\mathrm{ln}\:{b}}}=\frac{\mathrm{ln}\:{n}}{\mathrm{2ln}\:{b}}\:\Rightarrow\:\mathrm{ln}\:{n}\:=\mathrm{4ln}\:{b} \\ $$$$\frac{{b}\mathrm{ln}\:{n}}{\mathrm{ln}\:{b}}=\frac{\mathrm{ln}\:{bn}}{\mathrm{ln}\:{b}}\:\Rightarrow\:\mathrm{ln}\:{n}\:=\frac{\mathrm{ln}\:{b}}{{b}−\mathrm{1}} \\ $$$$\Rightarrow \\ $$$$\mathrm{4ln}\:{b}\:=\frac{\mathrm{ln}\:{b}}{{b}−\mathrm{1}}\:\Rightarrow\:{b}=\frac{\mathrm{5}}{\mathrm{4}}\:\Rightarrow\:{n}=\frac{\mathrm{625}}{\mathrm{256}} \\ $$…

Question-191395

Question Number 191395 by Mingma last updated on 23/Apr/23 Answered by Rasheed.Sindhi last updated on 23/Apr/23 $${xy}+{yz}+{zx}=\mathrm{0};\:\:\frac{{y}+{z}}{{x}}+\frac{{x}+{z}}{{y}}+\frac{{x}+{y}}{{z}}=? \\ $$$$\frac{{y}+{z}}{{x}}+\frac{{x}+{z}}{{y}}+\frac{{x}+{y}}{{z}} \\ $$$$=\frac{{y}+{z}}{{x}}+\mathrm{1}+\frac{{x}+{z}}{{y}}+\mathrm{1}+\frac{{x}+{y}}{{z}}+\mathrm{1}−\mathrm{3} \\ $$$$=\frac{{x}+{y}+{z}}{{x}}+\frac{{x}+{y}+{z}}{{y}}+\frac{{x}+{y}+{z}}{{z}}−\mathrm{3} \\ $$$$=\left({x}+{y}+{z}\right)\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\right)−\mathrm{3}…

Question-191394

Question Number 191394 by Mingma last updated on 23/Apr/23 Answered by Rasheed.Sindhi last updated on 23/Apr/23 $$\sqrt{{a}}\:+\frac{\mathrm{2022}}{\:\sqrt{{b}}\:}=\mathrm{2023} \\ $$$$\mathrm{2023}\sqrt{{b}}\:−\sqrt{{a}}\:\sqrt{{b}}\:=\mathrm{2022} \\ $$$$\sqrt{{b}}\:\left(\mathrm{2023}−\sqrt{{a}}\:\right)=\mathrm{2}×\mathrm{3}×\mathrm{337} \\ $$$$\begin{cases}{\sqrt{{b}}\:=\mathrm{2}\:\wedge\:\mathrm{2023}−\sqrt{{a}}\:=\mathrm{3}×\mathrm{337}}\\{\sqrt{{b}}\:=\mathrm{3}\:\wedge\:\mathrm{2023}−\sqrt{{a}}\:=\mathrm{2}×\mathrm{337}}\\{\sqrt{{b}}=\mathrm{337}\:\wedge\:\mathrm{2023}−\sqrt{{a}}\:=\mathrm{2}×\mathrm{3}\:}\\{\sqrt{{b}}\:=\mathrm{2}×\mathrm{3}\:\wedge\:\mathrm{2023}−\sqrt{{a}}\:=\mathrm{337}}\\{\sqrt{{b}}\:=\mathrm{2}×\mathrm{337}\:\wedge\:\mathrm{2023}−\sqrt{{a}}\:=\mathrm{3}}\\{\sqrt{{b}}\:=\mathrm{3}×\mathrm{337}\:\wedge\:\mathrm{2023}−\sqrt{{a}}\:=\mathrm{2}}\\{\sqrt{{b}}\:=\mathrm{2}×\mathrm{3}×\mathrm{337}\:\wedge\:\mathrm{2023}−\sqrt{{a}}\:=\mathrm{1}}\end{cases}\: \\ $$$$\begin{cases}{{b}\:=\mathrm{4}\:\wedge\:{a}\:=\left(\mathrm{2023}−\mathrm{3}×\mathrm{337}\right)^{\mathrm{2}}…

1-4-1-2-7-7-1-3-2-4-7-10-1-3-5-2-4-6-7-

Question Number 125857 by Dwaipayan Shikari last updated on 14/Dec/20 $$\mathrm{1}+\mathrm{4}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{7}} +\mathrm{7}\left(\frac{\mathrm{1}.\mathrm{3}}{\mathrm{2}.\mathrm{4}}\right)^{\mathrm{7}} +\mathrm{10}\left(\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}}{\mathrm{2}.\mathrm{4}.\mathrm{6}}\right)^{\mathrm{7}} +… \\ $$ Answered by Olaf last updated on 14/Dec/20 $$\mathrm{S}\:=\:\mathrm{1}+\underset{{n}=\mathrm{1}} {\overset{\infty}…