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 his 65th birthday, we need to calculate the accumulated value of the investments over all the years.
First, let's calculate the investment amount for each year. The student starts with $100 on his 21st birthday and increases the investment by $100 each subsequent year. This creates an arithmetic gradient of $100.
Let's break down the problem into smaller steps:
Step 1: Calculate the number of years the investments will grow for.
The student plans to invest from his 21st birthday to his 55th birthday. So, the investments will grow for a total of 55 - 21 + 1 = 35 years.
Step 2: Calculate the accumulated value of the investments using the compound interest formula.
The compound interest formula is given by: A = P(1 + r/n)^(nt)
Where:
A = Accumulated value
P = Principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, the principal amount (P) will vary each year due to the arithmetic gradient. So, we'll need to calculate the accumulated value for each year separately and sum them up.
Step 3: Calculate the accumulated value for each year.
Starting with the 21st birthday, we'll calculate the accumulated value year by year.
Year 21:
Principal amount (P) = $100
Interest rate (r) = 12% or 0.12 (as a decimal)
Number of times compounded per year (n) = 1
Number of years (t) = 35
Accumulated value for Year 21:
A = $100(1 + 0.12/1)^(1*35)
Step 4: Repeat Step 3 for each subsequent year until the 55th birthday.
Year 22:
Principal amount (P) = $200 (previous year's accumulated value + $100 gradient)
Interest rate (r) = 12%
Number of times compounded per year (n) = 1
Number of years (t) = 35 - 1 = 34 (as of the 22nd birthday)
Accumulated value for Year 22:
A = $200(1 + 0.12/1)^(1*34)
Repeat this process for each subsequent year, modifying the principal amount and the number of years accordingly.
Step 5: Sum up all the accumulated values calculated in Step 4.
Once you have calculated the accumulated values for each year, sum them up to find the future worth of the investments on the student's 65th birthday.