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 determine the future worth of the investments on the student's 65th birthday, we need to calculate the total investment made from his 21st to 55th birthday, and then calculate the future value of that investment at a 12% rate of return for the additional 10 years from his 55th to 65th birthday.
Let's break down the problem step by step:
Step 1: Calculate the total investment made from his 21st to 55th birthday.
The investment made on the 21st birthday is $100.
Each subsequent year, the investment increases on a $100 arithmetic gradient.
We can use the formula for the sum of an arithmetic series to calculate this total investment:
Sum = (n/2)(2a + (n - 1)d)
Where:
n = number of terms (number of years from 21st to 55th birthday = 55 - 21 + 1 = 35)
a = first term (investment on the 21st birthday = $100)
d = common difference (arithmetic gradient = $100)
Sum = (35/2)(2(100) + (35 - 1)(100))
Sum = (35/2)(200 + 34(100))
Sum = (35/2)(200 + 3400)
Sum = (35/2)(3600)
Sum = 35 * 1800
Sum = $63,000
Therefore, the total investment made from his 21st to 55th birthday is $63,000.
Step 2: Calculate the future value of the $63,000 investment for the additional 10 years from his 55th to 65th birthday.
We can use the formula for the future value of a single cash flow compounded annually to calculate this future value:
FV = PV(1 + r)^n
Where:
FV = future value (what we want to find)
PV = present value (the $63,000 total investment made from his 21st to 55th birthday)
r = annual interest rate (12% or 0.12 as a decimal)
n = number of compounding periods (10 years)
FV = 63,000(1 + 0.12)^10
FV = $63,000(1.12)^10
FV = $63,000(1.4049)
FV = $88,376.70 (rounded to two decimal places)
Therefore, the future worth of the investments on his 65th birthday is approximately $88,376.70.
To solve this problem, we need to calculate the future worth of the investments on the student's 65th birthday.
Let's break down the problem step by step:
1. Determine the initial investment: The student set aside $100 on his 21st birthday. So, this amount will be our starting point.
2. Calculate the annual increment: Each subsequent year, starting from the 22nd birthday, the student plans to increase the investment on a $100 arithmetic gradient. This means that each year, the investment amount will increase by $100.
3. Calculate the number of years of investment: The student plans to invest until his 55th birthday. So, he will be investing for a total of (55 - 21) = 34 years.
4. Calculate the growth rate: The problem states that the student can achieve a 12% rate of return. This rate of return will be expressed as a decimal by dividing 12 by 100, giving us 0.12.
Now, let's use these steps to set up the formula for finding the future worth of the investments:
FV = PV(A/G, r, n) + (A/G, r, m)
Where:
FV = Future Value
PV = Present Value (initial investment)
A = Annual Increment on the gradient
G = Number of years on the gradient
r = Interest rate (as a decimal)
n = Number of years of investment after the gradient ends
m = Number of years of investment on the gradient
In this case, we have:
PV = $100
A = $100
G = 1 (since the increment is applied annually)
r = 0.12 (12% as a decimal)
n = 10 (from the 55th birthday to the 65th birthday)
m = 30 (from the 22nd birthday to the 55th birthday)
Now, let's substitute the values into the formula:
FV = $100(A/G, 0.12, 30) + (A/G, 0.12, 10)
Calculating each part separately:
(A/G, 0.12, 30) = (30 * $100) * [((1 + 0.12)^30 - 1) / 0.12]
(A/G, 0.12, 10) = (10 * $100) * [((1 + 0.12)^10 - 1) / 0.12]
Finally, substitute the calculated values back into the main formula to find the future worth of the investments on the student's 65th birthday.