Formula for calculating the total interest payable over the life of a loan with linear redemption scheme I am trying to calculate the lifetime interest paid on a loan with a linear redemption scheme. I know that I can enter it into a spreadsheet and take the sum of the interest, but is there a formula to calculate such a value when you know the balance, duration, monthly redemption and the interest rate? I've done a lot of searching, but end op mostly at annuity calculations or loans with a fixed repayment.
P: Principal (amount) of loan
R: The monthly redemption on the loan
T: The term of the loan (i.e. the number of repayments)
r: The annual interest rate
If for example the Principal is $120.000$, the monthly redemption $1.000$, the term 120 months and the nominal yearly interest is $3%$ paid monthly at the end of each month.
Then the interest is $120.000*(0,03/12)$ the first month. The second month it's $(120.000-1000)*(0,03/12)$. The third it's $(120.000-2*1000)*(0,03/12)$ and so on. 
What would be the formula to sum this whole series up? Thank you in advance.
 A: Clearly the total principal payments are $120000$. 
The total interest payments are:
\begin{align*}
120000 &\times \frac{0.03}{12} + (120000-1000) \times \frac{0.03}{12}+(120000-2000)\times \frac{0.03}{12}+\cdots + 1000 \times \frac{0.03}{12}\\
&= \frac{0.03}{12}\times\left(\sum_{k=1}^{120} 1000\times k\right)\\
&= \frac{0.03}{12}\times 1000 \times \sum_{k=1}^{120}k \\
&= \frac{0.03}{12} \times 1000\times \frac{120\times 121}{2} \\
&= 18150.
\end{align*}

Edit 1: In general, the total interest payments will be:
$$\frac{r}{12}\sum_{k=1}^T Rk = \frac{r}{12}R\frac{T(T+1)}{2}= \frac{rP(T+1)}{24}$$
(since $P=RT$ - this assumes the loan is repaid in full after $T$ terms). 

Edit 2: If the loan is not paid in full after $T$ terms (i.e., $RT<P$), then the interest in the first $T$ terms is:
\begin{align*}
\frac{r}{12}&\left(P + (P-R) + (P-2R) + \cdots + (P-(T-1)R)\right)\\
&=\frac{r}{12} \sum_{k=0}^{T-1}(P-Rk)\\
&= \frac{r}{12}\left(\sum_{k=0}^{T-1} P  - R \sum_{k=0}^{T-1}k \right)\\
&= \frac{r}{12}\left(TP - R \frac{(T-1)T}{2}\right)\\
&= \frac{rT}{12} \left(P- \frac{R(T-1)}{2}\right).
\end{align*}
You can check that when $RT=P$, this expression coincides with the original answer.
