How many months will it take to accumulate $1,000,000 if you have $5,000 today and save $1,500 per month at an annual interest rate of 5.5%?

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To determine how many months it will take to accumulate $1,000,000 starting with $5,000 and saving $1,500 each month at an annual interest rate of 5.5%, we can utilize the future value of an annuity formula combined with the future value of a lump sum investment.

First, the future value of a lump sum (the initial $5,000) can be computed using the formula:

[ FV = P(1 + r/n)^{nt} ]

Where:

( FV ) = future value of the investment
( P ) = principal amount (initial investment)
( r ) = annual interest rate (as a decimal)
( n ) = number of times interest is compounded per year
( t ) = time in years

For the monthly savings, we use the future value of an annuity formula:

[ FV = PMT \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) ]

Where:

( PMT ) = payment amount per period (monthly savings)

Setting up an equation that accounts for both the lump sum and the monthly savings allows