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Invest 1000 dollars at bank at 3 percent interest compound annually. Every year the bank deducts 15 dollars in charges.

If $A_n$ is the value of the investment at the end of $n$ years write down a difference equation and initial condition for $A_n$

Solve this difference equation and hence calculate the value of the investment at end of 10 years.

This is part of the solution given but I do not understand the part 15/0.03 can anyone explain?

\begin{equation} A_{n+1}=1.03A_{n-1}-15,A_{0}=$1000 \end{equation}

\begin{equation} A_{n} = A*
\end{equation}

\begin{equation} A*=1.03A*-15,A*=15/0.03=500 \end{equation}

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1 Answer 1

up vote 2 down vote accepted

In your first equation, the difference of the subscripts should be 1, not 2. If $A*$ is the limit, you are seeking a value of the account that doesn't change. From $A*=1.03A*-15,$ you add $15$ to each side and subtract $A*$ from each side to get $15=0.03A*$, then divide by $0.03$ to find $A*$. But this doesn't get the value after 10 years, it gets the value that would be stable. In fact, the account is growing. The first year you get 30 in interest and pay 15 in fees, so $A_1=1015$ and it will grow by more than \$15 each year thereafter.

Added: as this is a linear recurrence, the full solution is a particular solution to the full equation plus any solution to the homogeneous part. You have found $A*=500$ regardless of $n$ is a solution to the whole recurrence without the initial condition. Now the homogeneous part $A_{n+1}=1.03A_n$ has the solution $A_n=a1.03^n$ for any constant $a$, so the general solution is $A_n=500+a1.03^n$ Given that $A_0=1000$, we can solve to find $a=500$ and the full solution is $A_n=500+500\cdot1.03^n$

Added 2: You could check out Wikipedia, though it may be too complex. Your basic equation looks like $A_{n+1}=rA_n+C$, which is called a linear recurrence because none of the $A$'s are multiplied by each other-they can be multiplied by constants or functions of $n$ and the equation is still linear. It is convenient to also consider the homogeneous part of the equation, with all terms that do not involve $A$'s, which here is $A_{n+1}=rA_n.$ The point is that if you add any solution to the homogeneous equation to a solution of the original equation, it is again a solution. Let us suppose $f(n)$ satisfies the original equation, so $f(n+1)=rf(n)+c$, and $g(n)$ satisfies the homogeneous part, so $g(n+1)=rg(n)$. Also any constant times $g(n)$ satisfies the homogeneous part. Then $f(n+1)+kg(n+1)=rf(n)+c+rkg(n)=r(f(n)+kg(n))+c$. So to find all solutions to the original equation, we can find any solution (called a particular solution) to the original equation and add to it all solutions of the homogeneous equation. An easy particular solution to find is to assume it does not depend on $n$, which is your $A*$. If we just substitute $f$ (instead of $f(n)$) into the original equation, we find $f=\frac{c}{1-r}$, which is where your $A*=\frac{-15}{-.03}=500$ came from-it is the constant solution to your equation. To find all solutions to the homogeneous equation, we have $A_{n+1}=rA_n$. The solutions can be found be inspection and are $dr^n$ for some constant $d$. So the general solution is $A_{n}=dr^n+\frac{c}{1-r}=d1.03^n+500$. To find $d$, you use your initial condition. Here we are given that $A_0=1000=d1.03^n+500=d+500,$ so $d=500$ and the full solution is $A_n=500+500\cdot1.03^n$

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if 15 is a constant C instead? –  optimus Apr 20 '11 at 14:52
    
I don't see $C$ in the problem, but it could be in your examples of solving recurrences. –  Ross Millikan Apr 20 '11 at 14:53
    
If the amount of fees to be deduct is replace as C. Particular solution: a* = 1.03a* - C Homogeneous solution: ? I am pretty weak in the concepts of recurrence relation. –  optimus Apr 21 '11 at 10:43
    
appreciate you can explain the overall concepts, and techniques to answer this type of question. dont understand the lecture notes –  optimus Apr 21 '11 at 10:46
    
@liangteh: see if this helps –  Ross Millikan Apr 21 '11 at 13:17

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