How can we use induction on 'i' here to form first to second equation?

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This is for a multi-class M/M/1/∞/PR queue. The equations in the talk are 3.40 and 3.41 where it is given:

$$\sigma_i \equiv\sum_{k=1}^i \rho_k,i=1,2,\cdots,r $$

solution to equation