A 20-year-old student decided to set aside $100 on
his 21st birthday for investment. Each subsequent
year through his 55th birthday, he plans to increase
the investment on a $100 arithmetic gradient. He will
not set aside additional money after his 55th birth-
day. If the student can achieve a 12% rate of return,
what is the future worth of the investments on his
65th birthday?
This is what I have so far:
100(A/G,12%,30)+(A/G,12%,10)
I do not understand how to set up this problem!!! Help!!!
To solve this problem, we need to find the future worth of the investments on the student's 65th birthday using the given information. Let's break it down step by step.
First, let's understand the concept of an arithmetic gradient. An arithmetic gradient refers to a series of increasing (or decreasing) payments, in which the difference between each payment is constant. In this case, the student plans to increase his investment by $100 each year.
Now, let's define some terms:
- A/G: The future worth of an arithmetic gradient.
- 12% rate of return: This means that the investment will grow at an annual interest rate of 12%.
To solve the problem, we can use the formula for the future worth of an arithmetic gradient:
A/G = (R * (1 + i)^n - i) / i^2
Where:
- A/G: Future worth of the arithmetic gradient
- R: Initial payment
- i: Interest rate per period (in this case, 12% or 0.12)
- n: Total number of periods
Now, let's apply this formula to the given problem:
Step 1: Calculate the A/G value from the 21st to the 55th birthday:
A/G1 = (100 * (1 + 0.12)^35 - 0.12) / 0.12^2
Step 2: Calculate the A/G value from the 56th to the 65th birthday:
A/G2 = (100 * (1 + 0.12)^9 - 0.12) / 0.12^2
Step 3: Calculate the future worth of the investments on the 65th birthday:
Future worth = 100 * A/G1 + A/G2
Now, you can substitute the values into the formula and solve for the future worth.