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Math Help - Forming an equation, then finding when last $400 is received

  1. #1
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    Forming an equation, then finding when last $400 is received

    Mrs Wong retired in 2006, she put a sum of $5000 into a fund that has a constant rate of return of 5 % per annum. Starting in 2006, she withdraws $400 each year and gives the money to her granddaughter as a birthday gift.

    a) Denote the amount of money Mrs Wong has at time t years by $x.

    b) In which year will the granddaughter receive her last $400?

    a) x =5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t

    b) In the year granddaughter receives last $400, x<400

    5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t<400
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  2. #2
    Grand Panjandrum
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    Re: Forming an equation, then finding when last $400 is received

    Quote Originally Posted by Punch View Post
    Mrs Wong retired in 2006, she put a sum of $5000 into a fund that has a constant rate of return of 5 % per annum. Starting in 2006, she withdraws $400 each year and gives the money to her granddaughter as a birthday gift.

    a) Denote the amount of money Mrs Wong has at time t years by $x.

    b) In which year will the granddaughter receive her last $400?

    a) x =5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t

    b) In the year granddaughter receives last $400, x<400

    5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t<400
    Part (a) is wrong.

    x_0=5000; (or 4600 depending on the interpretation of the wording of the question)

    x_1=x_0(1+0.05)-400

    x+2=(x_0(1+0.05)-400)(1+0.05)-400=x_0(1+0.05)^2-400(1+(1+0.05)

    :
    :

    x_k=x_0(1+0.05)^k-400[1+(1+0.05)+...+(1+0.05)^{k-1}]\\ \\ \phantom{SSSS}=x_0(1+0.05)^k-400\frac{1-(1+0.05)^k}{1-(1+0.05)}

    CB
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