Required initial investment at the beginning of the first year if the fund earns 6% compounded annually?

Question

A businessman wants to withdraw $3,000 (including capital) from an investment fund at the end of each year for 5 years. How should he compute his required initial investment at the beginning of the first year if the fund earns 6% compounded annually?

Answer 1

The easiest way to calculate this, is to calculate each withdrawal and the interest individually. There are 5 withdrawals.

Step 1: Write the equation:
The total amount should be $15000.
At the end of the fourth year, there needs to be enough principal + interest to have $3000:
3000 = p1 * 1.06
p1 = $2830.19

At the end of the third year, there needs to be enough principal + interest to have 3000 + the principal above:
3000 + p1 = p2 * 1.06
p2 = 5830.19/1.06 = $5500.18

And so on:
p3 = (3000 + p2) / 1.06 = 8019.035848
p4 = (3000 + p3) / 1.06 = 10395.31684
p5 = (3000 + p4) / 1.06 = 12637.09136 (solution!)

Answer 2

in case you ever took precalculus or another math type which covers homes of applications, you study a thank you to be certain the area of the function. between the pink flags to look out for is once you divide by skill of 0. on your expression, whilst x = 3, the denominator is 0. to that end, the area is each and every selection beside 3. regrettably, the selection you attempt to plug in (x = 3) is the only selection that would not artwork in this function. to work out this standard hand, you0 can graph this function on a TI-80 3. in case you zoom in close to the graph at x = 3, you will see that there is a sparkling spot there! it extremely is with the help of the fact, as pronounced above, there in simple terms isn't a value of the expression at x = 3. you would be able to declare, nicely it feels like the respond could desire to be 6, watching the graph. this concept of what the respond "could desire to be" is what limits are all approximately. The values of the function on the left and proper of x = 3 all bypass in direction of 6 as you get closer and closer. So we are saying the cut back as x is going to 3 is 6. So despite the fact that that's no longer technically the respond, 6 is your suitable selection. 0 heavily isn't marvelous in any sense. the very suitable answer is to declare that the expression is undefined at x = 3. This concern illustrates why 0/0 is named indeterminate. in this concern, 0/0 in a fashion equals 6. the concept that 0/0 can equivalent something is quite the essence of calculus.