In my last post, I developed a recursion formula for compound interest where the interest is compounded annually:

*A*_{n+1} = *A*_{n} + *A*_{n} × *r*/100 where you use the percentage value of *r* or*A*_{n+1} = *A*_{n} + *A*_{n} × *r* where the decimal equivalent of *r* is used.

These are good formulas if you want to know how much money you will have next year, but will take a while if you want to know how much you will have in 10 years. However, we can use the recursion formula to help develop the direct formula.

I will use the decimal equivalent equation so I don’t have to keep writing “/100”. I will add that at the end of this development.

Using the recursion formula, at the end of year 1 you will have

*A*_{1} = *A*_{0} + *A*_{0} × *r* = *A*_{0}(1+r)

Please review my post on factoring, specifically the distributive property, if you do not understand why *A*_{0} + *A*_{0} × *r* = *A*_{0}(1+r).

At the end of year 2, you will have *A*_{2} = *A*_{1} + *A*_{1} × *r* = *A*_{1}(1+r). But we have *A*_{1} in terms of our initial investment *A*_{0} above. So substituting *A*_{1} = *A*_{0}(1+r) into *A*_{2} = *A*_{1}(1+r) results in

*A*_{2} = *A*_{1}(1+r) = *A*_{0}(1+r)(1+r) = *A*_{0}(1+r)²

Wait! The exponent 2 of (1+r) is the same as the current period (the subscript of *A*_{2}. Wouldn’t it be nice if this pattern continued? Let’s do one more iteration, substituting *A*_{2} with the above equation:

*A*_{3} = *A*_{2} + *A*_{2} × *r* = *A*_{2}(1+r) = *A*_{0}(1+r)²(1+r) = *A*_{0}(1+r)³

Wow! It looks like this is true. In fact, it is. The exponent of the (1 + *r*) part is the same as the period of interest. If you want to find how much you will have after 10 years, you can use the direct formula:

*A*_{n} = *A*_{0}(1 + *r*)^{n}

So in our example of investing $1000 at 3% interest per annum (that is each year), compounded annually, you will have after 10 years

*A*_{10} = *A*_{0}(1 + *r*)^{10} = 1000(1 + 0.03)^{10} = $1343.92

Of course, you need a calculator that can do exponents, or you can use the all-knowing internet. The percentage form of the direct equation is

*A*_{n} = *A*_{0}(1 + *r*/100)^{n}

where you would substitute the “3” in for *r* instead of “0.03”.

I said last time that interest can be applied (compounded) more frequently than annually. I will continue this example next time where we do just that.