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Find the value of $$x$$ while $\sqrt{x+\sqrt{x^{2}+\sqrt{x^{3}+\sqrt{x^{4}+...}}}}=\sqrt{2x+1}$

Note by Jason Snow
10 months ago

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Can you write some terms of the series in LHS ? · 8 months, 2 weeks ago

Do you know the answer · 10 months ago

Desmos says that $$x=4$$ but I'm not sure if it's correct · 10 months ago

$$(n+1)=\sqrt { n^{ 2 }+2n+1 } =\sqrt { 1+\sqrt { n^{ 4 }+4n^{ 3 }+4n^{ 2 } } } =\sqrt { 1+\sqrt { 4{ n }^{ 3 }+\sqrt { n^{6}+8n^6+16n^4}}}$$

continuing the procedure we get

$$\sqrt { 1+\sqrt { 4n^{ 3 }+\sqrt { 16n^{ 4 }+\sqrt { 64{ n }^{ 12 }... } } } }$$

for n=1 we have

$$\sqrt { 1+\sqrt { 4+\sqrt { 16+\sqrt { 64... } } } }$$

squaring then subtracting one from each side we have above equation when $$x=4$$

we have the expression =3 when x=4

the other equation is also 3 when x=4

hence proved · 9 months, 4 weeks ago

Thanks !! :D · 9 months, 4 weeks ago

desmos is right

i'll show you why,it will take a bit of time to type it in LaTeX · 9 months, 4 weeks ago