Does it have any value?

Try if you can find value of below expression: \[\sqrt{1+\sqrt{\sqrt{2+\sqrt{\sqrt{\sqrt{3+\sqrt{\sqrt{\sqrt{\sqrt{4+\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{5...}}}}}}}}}}}}}}}=?\]

Any solution will be appreciated

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
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Comments

@Yajat Shamji, @Jeff Giff, @Mahdi Raza, @Aryan Sanghi, @Vinayak Srivastava, @Barry Leung, @Kumudesh Ghosh, @Alak Bhattacharya

Zakir Husain - 11 months, 3 weeks ago

I think it is almost equal to $1.53$

Above expression can be written as $\sqrt{1 + ^4\sqrt{2 + ^8\sqrt{3 + ....}}}$

So, the r $^{th}$ root is $2^r$ which is increasing exponentially whereas the value inside root is increasing linearly. So, it will be equal to $1$ after each root is simplified, so finally it will come as $\boxed{\sqrt{1 + 1.4} \approx 1.53}$ . Hope it helps.

Aryan Sanghi - 11 months, 3 weeks ago

even $\sqrt{1+\sqrt{\sqrt{2+\sqrt{\sqrt{\sqrt{3}}}}}}>\sqrt{2}$

Zakir Husain - 11 months, 3 weeks ago

I know, I mean it will converge near 1.53 approx.

Aryan Sanghi - 11 months, 3 weeks ago

@Aryan Sanghi – Please explain your answer.

A Former Brilliant Member - 11 months, 3 weeks ago

@A Former Brilliant Member – Actually I'll try but I guess I can't explain better. @Zakir Husain can you please help?

Aryan Sanghi - 11 months, 3 weeks ago

@Aryan Sanghi – I myself don't have any idea, it came to my mind and just stuck there. That's why I have wrote this note.

Zakir Husain - 11 months, 3 weeks ago

@Aryan Sanghi – Please

A Former Brilliant Member - 11 months, 3 weeks ago

@A Former Brilliant Member – See, the rth has its root value $2^r$ , isn't it? Now, the number inside the root is only $r$ . So, we can see that value of root is increasing exponentially whereas the number inside root is increasing linearly. So, the root will take it closer and closer to 1 as we move right.

So, we could ignore values more than 3. So, answer is $\sqrt{1 + ^4\sqrt{2 + ^8\sqrt{3}}} \approx 1.53$

Hope I explained well.

Aryan Sanghi - 11 months, 3 weeks ago

@Aryan Sanghi – Oooo,i've understood.I thought that you have done some substitution.Nice method of approximaton..are you preaparing for jee mains???

A Former Brilliant Member - 11 months, 3 weeks ago

@A Former Brilliant Member – Yes, for JEE mains and advanced.

Aryan Sanghi - 11 months, 3 weeks ago

@Aryan Sanghi – Nice

A Former Brilliant Member - 11 months, 3 weeks ago

Is it correct now @Zakir Husain, I have edited the solution.

Aryan Sanghi - 11 months, 3 weeks ago

@Aryan Sanghi – It is $\approx1.53...$

Zakir Husain - 11 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$