You purchase a new house for \( \$ 240,000 \) today. If you live in that house for 25 years before selling it, assuming an inflation rate of \( 5 \% \), compounded continuously, what would be the value of the home when you decide to sell it?

To find the future value of the house after 25 years with continuous compounding at an inflation rate of 5%, you can use the formula for continuous compounding: \[ A = Pe^{rt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the rate of interest (in decimal). - \( t \) is the time the money is invested or borrowed for, in years. - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Substituting the known values: - \( P = 240,000 \) - \( r = 0.05 \) - \( t = 25 \) So: \[ A = 240,000 e^{(0.05 \times 25)} \] \[ A = 240,000 e^{1.25} \] \[ A \approx 240,000 \times 3.49034 \] \[ A \approx 836,081.41 \] Thus, the value of the home after 25 years would be approximately \( \$ 836,081.41 \). Looking to the past, the concept of continuous compounding has its roots in the work of mathematicians like Jacob Bernoulli and John Napier. The elegance of \( e \) lies not just in finance; it’s also prominent in natural growth processes, from biology to physics! To embrace the art of investing in property, remember that location, market trends, and property maintenance play pivotal roles in increasing your home's value. Do your research and avoid common pitfalls like neglecting regular upkeep or underestimating renovation costs!