# Gas station without pumps

## 2010 July 29

### Solving an infinite radical chain

Filed under: Uncategorized — gasstationwithoutpumps @ 20:23
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Pat’s Blog had a post about infinite radicals, which looked at $3=\sqrt{x+\sqrt{x + \sqrt{x +\cdots}}}$.  The problem is just as easy to solve for $x$ if you replace the 3 with $y$.

$y =\sqrt{x+\sqrt{x + \sqrt{x +\cdots}}}$

$y^2 = x + \sqrt{x+\sqrt{x + \sqrt{x +\cdots}}}$

$y^2 = x + y$

$x = y^2 - y$

Note that this technique is almost identical to how one solves an infinite geometric series, so would make a good challenge problem right after introducing geometric series.  This could be appropriate for a math team recreational puzzle.

One can obviously generalize to $n^{th}$ roots:

$y =\sqrt[n]{x+\sqrt[n]{x + \sqrt[n]{x +\cdots}}}$

$x = y^n - y$

And of course one would have to discuss when the formal manipulation makes sense.  For example, $y=1$ leads to the “solution” $x=0$, which is clearly wrong.  It might be worth some calculator time for the kids to play with different values of $x$ to see when the infinite radical makes sense. The value of $y$ for $x=1$ should be a familiar number.

One could also solve the quadratic to see what values $y$ takes on for different values of $x$, and look for a relationship between the parts of the quadratic formula and the solutions to the infinite radical chain.

1. Gas….,
The iteration of cube roots of 0 may “clearly” not be equal to one…. but the idea is suggestive in a limit sense.
Try the iterated cube roots with .1 or .05 etc… and presto, calculus happens..and of course, we get a discontinuity that makes it all even more interesting.

And the graph of x=y^3-y can be pretty interesting for them to try to explain..particularly the part of the curve where 0<y<1…

Comment by pat ballew — 2010 July 30 @ 06:52

• You’re right, of course. The function is continuous at 0, and has the value 1, as the formula suggests. Interestingly, the iteration can be applied to complex numbers as well, if one picks some definition of “the” square root (for example, real part >0 or real part =0 and imaginary part>=0). This allows extending below x<-0.25.

Incidentally, on my HP calculator, RPN notation allows the computation to take only 2 button presses per iteration, without any programming (for real values).

Comment by gasstationwithoutpumps — 2010 July 30 @ 17:31