I need help to derivate a formula for the present value of an annuity with varying payments with the first payment being P at time = 0, and add a constant Q every increment of time. $$P + (P+Q)v + (P+2Q)v^2+(P+3Q)v^3+ \dots + (P+(n-1)Q)v^{n-1}$$
A help would greatly appreciated.
Let your sum be $S$. Then $$S = \sum\limits_{k=0}^{n-1}\left(P+kQ\right)v^{k} = P\sum\limits_{k=0}^{n-1}v^{k} + Q \sum\limits_{k=0}^{n-1}kv^{k}.$$
Now, we have $$\sum\limits_{k=0}^{n-1}v^{k} = \frac{1-v^{n}}{1-v}$$ using the geometric series formula. Also from here, you can see that $$\sum\limits_{k=0}^{n-1}kv^{k} = \frac{v-nv^{n} + (n-1)v^{n+1}}{(1-v)^{2}}.$$
Putting everything together, we have
$$\begin{align} S &= P\cdot \frac{1-v^{n}}{1-v} + Q\cdot \frac{v-nv^{n} + (n-1)v^{n+1}}{(1-v)^{2}}. \end{align}$$
