For IIT aspirants 3

Algebra Level 3

let $x=\sqrt{7+ \sqrt{7- \sqrt{7+ \sqrt{7-....}}}}$ , then find out the value of $x!$

1 solution

Akshat Sharda
Nov 6, 2015

I made a generalization for this type of question :

$x=\sqrt{a+ \sqrt{a- \sqrt{a+ \sqrt{a-\ldots}}}}$

Here, $a$ must be $+ve$ .

The equation which I got as the result was :

$x^{2}-x-(a-1)=0\Rightarrow x=\frac{1±\sqrt{4a-3}}{2}$

Here, $a=7\Rightarrow x=-2,3.$

The answer must be +ve. Therefore, $x=3\Rightarrow x!=\boxed{6}$

nice solution, I also did the same thing

Atul Shivam - 5 years, 7 months ago

I read the factorial sign as an exclamation mark :(

A Former Brilliant Member - 5 years, 7 months ago

!!!!!! Lol !!!!!! It is mathematics instead of English :p :p

Atul Shivam - 5 years, 7 months ago

@Atul Shivam – Can someone explain how he got $x^2-x-(a-1)=0$ ?

Thanks

pickle lamborghini - 5 years, 7 months ago

$x= \sqrt{a+ \sqrt{a- \sqrt{a+ \sqrt{a-\ldots}}}}$

$y= \sqrt{a- \sqrt{a+ \sqrt{a-\sqrt{a+\ldots}}}}$

$x=\sqrt{a+y}$ and $y=\sqrt{a-x}$

After squaring and subtracting the above equations we get,

$x^2-y^2=x+y$

$x-y=1\Rightarrow y=x-1$

$\Rightarrow x^2=a+y\Rightarrow x^2=a+(x-1)$

$x^2-x-(a-1)=0$

Akshat Sharda - 5 years, 7 months ago

@Akshat Sharda – Wow that's really clever! That clears things up, thanks!

pickle lamborghini - 5 years, 7 months ago

@Pickle Lamborghini – No problem $\ddot \smile$

Akshat Sharda - 5 years, 7 months ago

I too did that at the first attempt.

Swapnil Das - 5 years, 7 months ago