# 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

Note by Zakir Husain
1 year ago

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- 1 year 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.

- 1 year ago

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

- 1 year ago

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

- 1 year ago

- 1 year ago

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

- 1 year ago

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.

- 1 year ago

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

Yes, for JEE mains and advanced.

- 1 year ago

Nice

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

- 1 year ago

It is $\approx1.53...$

- 1 year ago