I want to know how to apply a sequence of steps in a program with any mainstream programming language (like Ruby, Java, Clojure etc).
Find nominal interest rate given principal initial amount, future value amount, number of years allowed to reach target future amount along with monthly contribution at the end of each month. So, you could be having starting principal as 1000, future amount as 5000, allowed to get to this say in 3 years and you are contributing 100 at the end of each month? Similarly you could be contributing 1000 say at the end of each year? In both cases, how to get the interest rate with the contributions happening.
The rate is assumed to be compounding, that is compound interest applies here.
Best Answer
This is a problem with many parts, and it's difficult to tell from the question how much detail you need in each part.
The basic idea is that to write a formula that would tell you what the final value of the account would be, given the starting value, contribution at the end of each month, rate of interest earned per month, and number of months. The formula is given in https://math.stackexchange.com/a/395799; for payments made at the end of each month it is $$P=B(1+i)^n+A\dfrac{(1+i)^n-1}{i}$$ where $B$ is the initial value, $A$ is the deposit each month, $i$ is the interest rate, and $P$ is the final value of the account.
You are given the values of all the quantities in this formula except for $i$, which is the variable you must solve for. Unfortunately, there is no closed-form solution to find $i$, so instead you must rely on some kind of guess-and-refine method such as Newton's method. A good first guess for $i$ for most purposes would be to simply add up all the payments (which in your example comes to $4600$) and subtract this from the final value to find out how much interest the account had to earn, which in this example is $400$. Then estimate the average value of the account during the entire time; it's difficult to do this exactly without knowing the interest rate, but for the purpose of the first guess, try averaging the starting and ending values: $\frac12(1000 + 5000) = 3000$. Now solve for $i$ in $3000 (1 + 36i) = 400$. (The factor of $36$ is because we're looking for monthly interest, and there are $36$ months between the initial and final values of this account.) And remember, this is only the first guess; you must apply some further steps to refine the value until the formula above produces the correct value of $P$ (to within a reasonable degree of accuracy) given the other quantities.