# Forming an equation, then finding when last $400 is received Printable View • Dec 19th 2011, 09:19 PM Punch 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$• Dec 19th 2011, 10:35 PM CaptainBlack Re: Forming an equation, then finding when last$400 is received
Quote:

Originally Posted by 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