What is the formula for cumulative compound interest?

Question

I would like to start with a principal amount (P) in year 0, then add compound interest (C) to it for year 1, and then add that total value to the starting amount. So for example:

P=1000 C=2.5%

For three years growth, the outputs would be:

Year 0 1000.00
Year 1 2025.00    1000+(1000*1.025)
Year 2 3075.63    1000+(1000*1.025)+(1000*1.025*1.025)
Year 3 4152.52    1000+(1000*1.025)+(1000*1.025*1.025)+(1000*1.025*1.025*1.025)

What would the mathematical formula be for that please?

Answer 1

That would be the future value of an annuity.

You receive $\$1000$ each year, and you invest the proceeds at $2.5\%.$ How much do you have at the end of $n$ years?

$\sum_\limits{k=1}^n 1000(1.025)^k$

That is a geometric series and we have a formula for that.

$\sum_\limits{k=1}^n 1000(1.025)^k = 1000\frac {1.025(1.025^n-1)}{1.025-1}$

Answer 2

Given principal $P$ and interest rate $c>0$, at the end of year $t$ you have

$$P(1+(1+c)+...+(1+c)^t)=P\frac{(1+c)^{t+1}-1}{c}.$$