Find the value of \(x\) while \[\sqrt{x+\sqrt{x^{2}+\sqrt{x^{3}+\sqrt{x^{4}+...}}}}=\sqrt{2x+1}\]

Find the value of \(x\) while \[\sqrt{x+\sqrt{x^{2}+\sqrt{x^{3}+\sqrt{x^{4}+...}}}}=\sqrt{2x+1}\]

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TopNewestCan you write some terms of the series in LHS ? – Aditya Sky · 1 year ago

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Do you know the answer – Aniket Gupta · 1 year, 2 months ago

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– Jason Snow · 1 year, 2 months ago

Desmos says that \(x=4\) but I'm not sure if it's correctLog in to reply

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 – Hummus A · 1 year, 2 months ago

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– Jason Snow · 1 year, 2 months ago

Thanks !! :DLog in to reply

i'll show you why,it will take a bit of time to type it in LaTeX – Hummus A · 1 year, 2 months ago

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We have tried some methods that didn't seem to work, so looking forward to your solution :D – Jason Snow · 1 year, 2 months ago

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