Show these wavelengths are consistent with Rydberg formula

[Kara386]

Homework Statement

These wavelengths are emitted by a hot gas:
18.226, 13.501, 12.054 (in nanometres)

Show that they are consistent with the Balmer series for a hydrogen-like atom.
Which element do they correspond to?

Homework Equations

Rearranged Rydberg formula for hydrogen-like atoms:
$Z^2 = \frac{hc}{\lambda R}(\frac{1}{n^2}-\frac{1}{4})^{-1}$

The Attempt at a Solution

Z is nuclear charge, that's really what I need to find to identify the element, so I rearranged the Rydberg formula to get the above expression.

The only way I can think of to solve this is by trial and error. So I tried subbing in $n=3$ and the first wavelength, and found that that corresponded to $Z=6$, then $n=4$ and $n=5$ with the next two wavelengths respectively also gave $Z=6$. But I'm not sure that really satisfies the condition 'show that they are consistent', because I've just guessed n and it happened to work. Is there a better way to solve this?

 

[DrClaude]

Your approach is good.