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!!!
I think its
f= 100(P/G,12%,34)(F/G,12%,44)
To solve this problem, we need to find the future worth of the investments on the student's 65th birthday.
Let's break down the problem step by step to better understand how to set it up:
1. The student decides to set aside $100 on his 21st birthday for investment. This means he has 45 years to invest.
2. Each subsequent year, starting from his 22nd birthday, he plans to increase the investment by $100. This forms an arithmetic gradient, where each year's investment is $100 greater than the previous year.
3. The student will not set aside additional money after his 55th birthday. This means the last additional $100 will be added on the year he turns 55.
4. The student aims to achieve a 12% rate of return on his investments.
Now that we understand the problem, let's set it up systematically:
Step 1: Determine the number of terms in the arithmetic gradient.
The student starts investing at age 21 and stops at age 55. Therefore, there are 55 - 21 = 34 terms in the arithmetic gradient.
Step 2: Calculate the future worth of the investments for the first 34 terms.
To find the future worth of the investments for the first 34 terms, we can use the formula for the future value of an arithmetic gradient, which is:
Future Worth = G x (A/G, r, n) + P
where G is the common difference in each term of the gradient, A/G is the future worth of the gradient per dollar, r is the interest rate, n is the number of terms, and P is the initial principal.
In this case, the common difference is $100, the interest rate is 12%, and the number of terms is 34. The initial principal is $100, so we have:
Future Worth = 100 x (A/G, 12%, 34) + 100
Step 3: Calculate the future worth of the investments for the last 10 years.
Since the student will not set aside additional money after his 55th birthday, the future worth of the investments for the last 10 years will be the total value accumulated in the first 34 terms. Therefore, we can use the same formula as in Step 2:
Future Worth = 34 x (A/G, 12%, 34) + 100
Step 4: Calculate the final future worth.
To find the future worth of the investments on the student's 65th birthday, we need to add the future worth of the investments for the first 34 terms and the future worth of the investments for the last 10 years:
Final Future Worth = 100 x (A/G, 12%, 34) + 34 x (A/G, 12%, 34) + 100
Now you can substitute this equation into your original expression:
100(A/G, 12%, 30)+(A/G, 12%, 10)
The steps provided above should help you set up and solve the problem to find the future worth of the investments on the student's 65th birthday.