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
$$n(\lambda_{k + 1}) = n(\lambda_k) + \Delta\lambda_k \frac{dn(\lambda_k)}{d\lambda_k} + \text{ higher order}$$.