I start with a monthly contribution rate of $200$, which grows at the end of every year by $3\%$. I then know that the sum of all contributions after $40$ years would be $\frac{(200\cdot12)\cdot(1-1.03^{40})}{1-1.03}$ by using the formula for the sum of a geometric progression $\frac{a(1-r^n)}{(1-r)}$
I then add an interest rate to the total amount contributed until then, say $2\%$. So for example we would have:
Year 1: $200\cdot12=2400$
Year 2: $200\cdot12\cdot1.03+2400\cdot1.02=4920$
Year 3: $200\cdot12\cdot1.03^2+4920\cdot1.02=7564.6$
And so on.
I can easily calculate this for 40 years using excel.
My question is, is it possible to get the final sum using a simple formula (so no $\Pi,\Sigma$ sign used)?
I would need it in this fashion in order to plug it into a single excel cell.
Any help would be much appreciated.
At the start of year $k$, you make a contribution of $200\cdot1.03^k.$ (We start $k$ at $0$.) It then grows with interest for $40-k$ years, so at the end of year $39$ it has grown to $200\cdot1.03^k1.20^{40-k}$ and the total sum is $$200\cdot1.02^{40}\sum_{k=0}^{39}\left({1.03\over1.02}\right)^k$$
This is a geometric progression, which you already know how to sum.
EDIT
Oops, I missed the part about monthly contributions. It's not clear to me how the bank credits interest though. Your example makes it seem at though the bank credits interest on all amounts as though they had been contributed at the beginning of the year (most unusual!), in which case, you should just multiply the above answer by $12$.