I see similar questions but not exactly the answer I'm looking for. I need to reduce this formula down so I can see the steps and understand how it is arriving at the solution. Reason being is I need to write a program that will calculate this and not knowing how to solve it myself is a problem.
$$\frac{100000}{(1+r)^1} + \frac{150000}{(1+r)^2} + \frac{200000}{(1+r)^3} + \frac{250000}{(1+r)^4} + \frac{300000}{(1+r)^5} - 250000 = 0$$
The answer appears to be r = 56.73%
It looks like this answer is saying use another formula that will allow you to start trying different answers until you eventually get one right. What I'm looking for is to show how to reduce the formula step by step to see it remain r = 56.73%. Hopefully I explained that correctly.
Hint: First divide by 50000 and substitute $1/(1+r)=q$.
$$2q+3q^2+4q^3+5q^4+6q^5=5$$
Then note,
$S=2q+3q^2+4q^3+5q^4+6q^5$ and $qS=2q^2+3q^3+4q^4+5q^5+6q^6.$
Subtract both equations: $(1-q)S=2q+q^2+q^3+q^4+q^5-6q^6=q+q^2+q^3+q^4+q^5+q^6+q-7q^6.$
Now, use geometric series to sum $q+q^2+...+q^6$. Can you complete it from here?