How can this be an integer??

One of my friends told me that $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}}$ comes out to be an integer. Can anyone help me out with guessing that single digit integer?

#Algebra #NestedRadicals

Easy Math Editor

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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}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

The method to solve this kind of nested radical is to define a function $f(x)=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}$ Clearly $f(x)^2-1=xf(x+1)$ Now note that $f(0)=1$ We will use induction to show that $f(x)=x+1$ for all positive integers $x$ (which will in turn give us the value of $f(2)$ ) First note the base case $f(0)=1=0+1$ Now suppose $f(k)=k+1$ for some positive integer $k$ . Then $f(k)^2-1=kf(k+1)\implies (k+1)^2-1=kf(k+1)\implies k^2+2k=kf(k+1)\implies f(k+1)=k+2$ and the induction is done.

We conclude that $f(x)=x+1$ for all positive integers $x$ .

Now we compute the value of $f(2)$ simply as $2+1=3$ . :)

Nathan Ramesh - 6 years, 6 months ago

Thanks dude! Wikipedia had a more general one which was more difficult to be understood! Keep it up!

Pranjal Jain - 6 years, 6 months ago

Thanks! :)

There is still a slight issue with the induction though. (I think). See if you can find it.

Nathan Ramesh - 6 years, 6 months ago

@Nathan Ramesh – Yeah sure! I just checked till $f(x)^{2}-1=xf(x+1)$

Pranjal Jain - 6 years, 6 months ago

@Pranjal Jain – Here is the issue (which I'm not sure if it's really an issue, maybe @Calvin Lin can clear it up)

When going from $f(0)$ to $f(1)$ using the recurrence relation and solving for $f(1)$ , what do we get?

Nathan Ramesh - 6 years, 6 months ago

@Nathan Ramesh – Oh yeah! $\frac{0}{0}$ indeterminate form!!!

Pranjal Jain - 6 years, 6 months ago

Not quite. In essence, you have not proven anything as yet. You need the fact that $f(1) = 2$ to conclude that $f(k) = k+1$ .

The induction step is only valid if you can divide by $k$ (right at the end). Hence, this need not hold true for $k = 0$ . As such, your base case needs to be $k=1$ .

Calvin Lin Staff - 6 years, 6 months ago

@Calvin Lin does this prove the base case?

$2=\sqrt{1+1\cdot 3}$ $\sqrt{1+1\cdot 3}=\sqrt{1+1\sqrt{1+2\cdot 4}}$ $\sqrt{1+1\sqrt{1+2\cdot 4}}=\sqrt{1+1\sqrt{1+2\sqrt{1+3\cdot 5}}}$

We can continue this indefinitely

Nathan Ramesh - 6 years, 6 months ago

@Nathan Ramesh – This way you can directly prove mine one to be 3!!!!

$3=\sqrt{1+2×4}=\sqrt{1+2\sqrt{1+3×5}}$ and so on as you did in your solution!!

Pranjal Jain - 6 years, 6 months ago

@Pranjal Jain – Yes, of course.

Nathan Ramesh - 6 years, 6 months ago

Can you help me in proving this? I guess @Nathan Ramesh 's method is not bad and "something" can be done with induction.

Pranjal Jain - 6 years, 6 months ago

It is 3! And it is reported in http://en.wikipedia.org/wiki/Nested_radical

Chew-Seong Cheong - 6 years, 6 months ago

Is there any other possible approach?

Pranjal Jain - 6 years, 6 months ago

Not one that I know.

Chew-Seong Cheong - 6 years, 6 months ago

Its value is 3

This is very famous nested radical given by ramanujan.

Krishna Sharma - 6 years, 6 months ago

yes i also approached same question on this site, the key is ramanujan

Mardokay Mosazghi - 6 years, 6 months ago

From: $f(x)^2 -1 = xf(x+1)$ $( f(x)+1 ) ( f(x)-1 ) = xf(x+1)$

Suppose $f(x) = x+1$ , then: $(x+1+1)(x+1-1)=x(x+2)$ $x(x+2) \equiv x(x+2)$

It appears that $f(x)=x+1$ satisfies the first equation for all real values of $x$ . The special case of $x=0$ is also shown to be $f(0) = 0 +1$ .

I hope this is adequate.

Chew-Seong Cheong - 6 years, 6 months 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}`	$\boxed{123}$