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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 1 year, 11 months ago

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Can you write some terms of the series in LHS ?

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Do you know the answer

Desmos says that \(x=4\) but I'm not sure if it's correct

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

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

@Hummus A – Thanks !! :D

desmos is right

i'll show you why,it will take a bit of time to type it in LaTeX

We have tried some methods that didn't seem to work, so looking forward to your solution :D

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Easy Math Editor

`*italics*`

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italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

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Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

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`\sqrt{2}`

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`\boxed{123}`

## Comments

Sort by:

TopNewestCan you write some terms of the series in LHS ?

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Do you know the answer

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Desmos says that \(x=4\) but I'm not sure if it's correct

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

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

Log in to reply

Log in to reply

desmos is right

i'll show you why,it will take a bit of time to type it in LaTeX

Log in to reply

We have tried some methods that didn't seem to work, so looking forward to your solution :D

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