# An Earnest Challenge

There appears to be a lot of expertise amongst the members of this site regarding solving nested radicals, so I thought I'd share a challenging one for your radical enjoyment:

$f(x) = \displaystyle\sqrt{x + \sqrt{\frac{x}{2} + \sqrt{\frac{x}{4} + \sqrt{\frac{x}{8} + \sqrt{\frac{x}{16} + \sqrt{\frac{x}{32} + ......}}}}}}$.

The hope is that there is an exact solution, if only for $f(1)$ if not for $f(x)$ in general. I suppose one interesting feature of this function is that $(f(x))^{2} - x = f(\frac{x}{2})$. I'm sure that there are many more interesting features waiting to be discovered. Note by Brian Charlesworth
6 years, 6 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $\boxed{123}$

Sort by:

f(x) = 2sqrt(x)

- 6 years, 5 months ago

For higher degree of radicals f(x)= nth root of 2x

- 6 years, 5 months ago

Could it be written in this fashion?

$f(x) = \sqrt{\sum_{i=0} \frac{f(x)}{2^{i}}}$

- 6 years, 6 months ago

no.. I think the 'x' terms under summation are missing and 'i' should start from 1 instead of 0.. if I'm not wrong.

- 6 years, 5 months ago

It's a recursive function. I thought "i" should start from 0 since the first term is "x", not "x/2"

- 6 years, 5 months ago

Your problem does not have closed form but

$\sqrt{ x + \sqrt{\frac{x}{2} + \sqrt{\frac{x}{4} + \sqrt{\frac{x}{16} + \sqrt{\frac{x}{256} + \ldots}}}}}$

Can have a closed form

- 6 years, 6 months ago

It does? Cool. I'll have to figure out what that is, then.

- 6 years, 6 months ago

I was solving your radical and did a mistake and I solved the above radical :p, now I will post a problem on this ;)

- 6 years, 6 months ago

Haha. Well, a lot of "mistakes" have led to interesting discoveries. Ill keep an eye out for your problem.

- 6 years, 6 months ago

My bet is that there isn't any closed form expression for this, not even in the special case of $x=1$.

- 6 years, 6 months ago

You're probably right, but I'm getting used to seeing rabbits being pulled out of hats so I thought I'd post the problem just in case.

- 6 years, 6 months ago