Final answer:
To find the account balance at the end of month 10 with a $100 monthly deposit and 6% annual interest compounded monthly, the future value of a series of annuities formula is applied to each deposit. The balance at the end of month 10 approximately calculates to be $1,040.05.
Step-by-step explanation:
To calculate the balance at the end of month 10 for an account that saves $100 monthly at a 6% annual interest rate compounded monthly, we use the future value of an annuity formula. For each deposit, the formula is FV = P × {[{(1 + r)^n} - 1} / r}, where P is the periodic payment, r is the monthly interest rate, and n is the number of periods.
The monthly interest rate (r) is 6% annual rate divided by 12 months, which is 0.5%. Translating this into decimal form, we get 0.005. And since there are 10 periods, n is equal to 10.
The formula reckons the future value (FV) for each deposit made at the end of each period, which is then summed to get the total balance. Applying the formula yields:
- First Deposit: $100 × {[{(1 + 0.005)^10} - 1} / 0.005} = $101.53
- Second Deposit: $100 × {[{(1 + 0.005)^9} - 1} / 0.005} = $100.75
- ... and so on, until the tenth deposit which has no time to earn interest and remains $100.
Adding up the future values of all 10 payments amounts to the total balance, which for this scenario will be approximately $1,040.05.