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I am curious to know If dust extinction were to vary as a power-law so that $A_{\lambda}$ proportionate to $\lambda^{\beta}$ , what value of beta would give rise to Rv =3.1 which is the average in diffuse ISM. I can see a plot but I wanted to see how to derive Rv=3.1 mathematically

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The value is $\beta \simeq -1$ (e.g. see Cardelli et al. 1988). The extinction would not be well fitted by a power law, except over a very narrow wavelength range.

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