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 arithmeticgradient. Hewill
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?
To calculate the future worth of the investments on the student's 65th birthday, we need to determine the amount invested each year and then calculate the compounded value of those investments over time.
First, let's break down the problem and find the amount invested each year:
1. The student starts by setting aside $100 on his 21st birthday.
2. Each subsequent year, he plans to increase the investment by $100 as an arithmetic gradient.
To find the amount invested each year, we can use the formula for the nth term of an arithmetic sequence:
An = A1 + (n - 1) * d
Where:
An is the nth term of the sequence,
A1 is the first term,
n is the number of terms, and
d is the common difference.
In this case, A1 = $100, n = (65 - 21) = 44 (since he invests from his 21st to 65th birthday), and d = $100.
Using the formula, we can find the amount invested each year:
A44 = $100 + (44 - 1) * $100
A44 = $100 + 43 * $100
A44 = $100 + $4,300
A44 = $4,400
Now that we know the amount invested each year, we can calculate the future worth of the investments on his 65th birthday.
To calculate the future value of the investments, we can use the formula for compound interest:
FV = PV * (1 + r)^n
Where:
FV is the future value,
PV is the present value (initial investment),
r is the interest rate (as a decimal),
and n is the number of compounding periods.
In this case, the initial investment (PV) is $4,400, the interest rate (r) is 12% or 0.12, and the number of compounding periods (n) is (65 - 21) = 44 (since he invested from his 21st to 65th birthday).
Using the formula, we can calculate the future worth:
FV = $4,400 * (1 + 0.12)^44
Calculating this gives us:
FV ≈ $4,400 * (1.12)^44
FV ≈ $4,400 * 36.948
FV ≈ $162,567.16
Therefore, the future worth of the investments on the student's 65th birthday is approximately $162,567.16.