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$\displaystyle =5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t $

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

$\displaystyle 5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t<400$

Re: Forming an equation, then finding when last $400 is received

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**Punch** 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$\displaystyle =5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t $

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

$\displaystyle 5000+\frac{(5000)(1-0.05^t)}{1-0.05}-400t<400$

Part (a) is wrong.

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

$\displaystyle x_1=x_0(1+0.05)-400$

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

:

:

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

CB