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Question Number 110204 by Algoritm last updated on 27/Aug/20

Commented by Algoritm last updated on 27/Aug/20

$$⧫\mathbf{prove}\mathbf{that}!$$

Commented by prakash jain last updated on 28/Aug/20

$${(2^n\cdot 4+1)}^2=4^{n+2}+2^{n+1}\cdot 4+1$$$$(2^n.4+1)=\sqrt{4^{n+2}+(2^{n+1}\cdot 4+1)}$$$$3=\sqrt{9}$$$$=\sqrt{4+(2^0.4+1)}$$$$=\sqrt{4+\sqrt{4^2+2^1.4+1}}$$$$=\sqrt{4+\sqrt{4^2+\sqrt{4^3+2^2\cdot 4+1}}}$$$$\mathrm{and}\mathrm{so}\mathrm{on}$$

Commented by Algoritm last updated on 27/Aug/20

$$\mathrm{sir}\sqrt{9}=\sqrt{4+(2\times 4+1)}=\sqrt{4+9}???$$

Commented by prakash jain last updated on 28/Aug/20

$$\mathrm{correct}.\mathrm{but}\mathrm{the}\mathrm{basic}\mathrm{idea}\mathrm{remains}$$$$\mathrm{the}\mathrm{same}.$$

Answered by Dwaipayan Shikari last updated on 27/Aug/20

$$3=\sqrt{4+\sqrt{4^2+9}}$$$$3=\sqrt{4+\sqrt{4^2+\sqrt{4^3+17}}}$$$$3=\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{4^4+33}}}}$$$$3=\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{4^4+\sqrt{4^5+65}}}}}$$$$3=\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{4^4+\sqrt{4^5+......}}}}}$$

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