You deposit $1,000 today and want to save $100 each month beginning one month from today. Your account earns a 5 percent annual interest rate. How long will it take you to accumulate $5,000?
A) 29.3 months
B) 35.7 months
C) 42.6 months
D) 52.1 months

Question

Quizplus · Accepted Answer

To solve this problem, we can use the future value of a series formula for the monthly deposits and add the future value of the single lump sum deposit. The future value of a series formula is $ FV = P \times \frac{(1 + r)^n - 1}{r} $, where $P$ is the monthly payment, $r$ is the monthly interest rate, and $n$ is the number of payments. The future value of a single sum is $ FV = P \times (1 + r)^n $.Given:- Monthly deposit ($P$) = $100- Annual interest rate = 5%, so the monthly interest rate ($r$) = 5%/12 = 0.0041667- Initial deposit = $1,000We want to find $n$ such that the total future value is $5,000.First, calculate the future value of the initial deposit separately, as it does not grow at the same rate as the monthly deposits due to its immediate investment.For the initial $1,000 deposit, the future value formula simplifies to $ FV = P \times (1 + r)^n $, but we'll incorporate this into our total at the end since it's a separate calculation from the monthly deposits.For the monthly deposits, we rearrange the future value of a series formula to solve for $n$, but given the complexity of solving for $n$ directly due to the nature of the equation (it involves logarithms), we typically use financial calculators or software for precise calculations.Using a financial calculator or software with the inputs: $P = 100$, $r = 0.0041667$, and a target $FV = 5000 - FV$ of the initial deposit, we find that it takes approximately 35.7 months to accumulate $5,000, including the growth of the initial $1,000 deposit and the monthly deposits of $100 each.This calculation assumes the interest compounds monthly and that all deposits are made at the end of each month. The exact time can vary slightly based on the method of calculation or rounding, but option B is the closest and correct choice based on the standard approach to solving these types of financial problems.

You Deposit $1,000 Today and Want to Save $100 Each