How to approximate this relation?

[KFC]

I got a relation as follow

\lambda_k = \frac{2 n(\lambda_k) L}{k}

where \lambda_k is a wavelength at mode k, k is integer, n is the index of reflection, L is a constant. I am trying to find the change of wavelength between two adjacent mode approximately, the answer will be

\Delta\lambda_k \approx \frac{\lambda_k^2}{2n_gL}

where
n_g = n(\lambda_k) - \left.\lambda_k\frac{dn}{d\lambda}\right|_{\lambda_k}

I have no idea how to achieve this. Please give me some hint. Thanks

 

[CompuChip]

I don't see exactly how to do it right away, but n_g looks an awful lot like a first-order Taylor expansion of n around \lambda_k. So maybe if you rewrite n(\lambda_k) = \frac{k \lambda_k}{2 L}
and then do some expansion of the left hand side for
n(\lambda_{k + 1}) = n(\lambda_k) + \Delta\lambda_k \frac{dn(\lambda_k)}{d\lambda_k} + \text{ higher order}.