Final answer:
By calculating the future values of each deposit separately and adding them together, it is determined that the total amount in the account at the end of year 10, with an interest rate of 10% compounded annually on deposits of $5,000 now, $7,000 in year 3, and $2,000 per year in years 4 through 10, will be more than $50,000.
Step-by-step explanation:
To answer the question regarding the future value of a series of different deposits into an account with an interest rate of 10% compounded annually, a cash flow diagram needs to be created, and the future values of each deposit must be calculated. First, let's calculate the future value for each separate deposit.
- $5,000 deposited now will compound for 10 years.
- $7,000 deposited at the end of year 3 will compound for 7 years.
- $2,000 per year deposited at the end of years 4 through 10 will compound individually for each year (6 years for the first deposit, 1 year for the last deposit).
The future value (FV) of a single sum can be calculated using the formula FV = PV * (1 + r)^n where PV is the present value, r is the annual interest rate (as a decimal), and n is the number of compounding periods.
The future value of the $5,000 deposit is $5,000 * (1 + 0.10)^10.
The future value of the $7,000 deposit in year 3 is $7,000 * (1 + 0.10)^7.
The future value of each of the annual deposits is calculated separately, according to the number of years each amount is compounded, and then summed together.
The sum of future values of all these deposits will provide the total amount in the account at the end of year 10. We will find that this amount is more than $50,000.