Annual deposits of $3150 are made into a bank account earning 4 % interest per year.
(a) What is the balance in the account right after the 15th deposit if interest is calculated annually?
(b) What is the balance in the account right after the 15th deposit if interest is calculated continuously?
To find the balance right after the 15th deposit, we need to calculate the accumulated balance, taking into account the annual interest and the frequency of deposits.
(a) If interest is calculated annually, we can calculate the balance by using the formula for the future value of an ordinary annuity.
The formula for the future value of an ordinary annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value of the annuity
P = Deposit amount made per year
r = Interest rate per year
n = Number of deposits
In this case, P = $3150, r = 4% = 0.04 (decimal), and n = 15.
Calculating the future value using the formula:
FV = $3150 * [(1 + 0.04)^15 - 1] / 0.04
FV = $3150 * [1.04^15 - 1] / 0.04
FV ≈ $3150 * [1.740272 - 1] / 0.04
FV ≈ $10,934.20
So, the balance in the account right after the 15th deposit, if interest is calculated annually, is approximately $10,934.20.
(b) If the interest is calculated continuously, we can use the formula for compound interest:
A = P * e^(rt)
Where:
A = Accumulated balance
P = Deposit amount made per year
r = Interest rate per year
t = Number of years
In this case, P = $3150, r = 4% = 0.04 (decimal), and t = 15.
Calculating the accumulated balance using the formula:
A = $3150 * e^(0.04 * 15)
A ≈ $3150 * e^0.6
A ≈ $3150 * 1.8221188
A ≈ $5751.564
So, the balance in the account right after the 15th deposit, if interest is calculated continuously, is approximately $5,751.56.