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Admittedly, I just finished physics and calculas but some of my more basic math skills escape me.

I'm looking for a formula that will give me a total compounded value after x number of weeks. So for example lets say I start off with an initial value of $1,000.00. And every week I make 10% of that value. The 10% would be added onto the initial value and the next week I would make 10% of that total value.

Week 1: 1,000.00 * .1 = 100.00 1,000.00 + 100.00 = 1,100

week 2: 1,100.00 * .1 = 110.00 1,100.00 + 110.00 etc..

Is there a particular formula for this type of compounded exponential growth, and if so, what is the name of it?

Scott
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1 Answers1

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After $1$ week you have the initial amount $A$ times $1.1$, so $A(1.1)$.

Each week the amount you have at the beginning of the week gets multiplied by $1.1$. So after $2$ weeks you have $A(1.1)(1.1)=A(1.1)^2$. After $3$ weeks you have $A(1.1)^2(1.1)=A(1.1)^3$. After $4$ weeks you have $A(1.1)^4$. After $n$ weeks you have $A(1.1)^n$.

The method generalizes. If the weekly interest rate is $4.5\%$, that is, $0.045$, and the initial amount you have (or owe) is $A$, then after $n$ weeks the amount you have (or owe) is $A(1.045)^n$.

Remark: Since you just finished calculus, let us look at the problem another way. After $1$ week we have an amount $Ae^{\ln 1.1}$. Each week gives us another multiplication by the same factor, so after $n$ weeks we have $Ae^{(\ln 1.1)n}$. This is a special case of the usual formula for exponential growth.

André Nicolas
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