Assume the economy has gotten better and that the funds now have a return of 7.72% compounded monthly. Diane salary has risen now she can contribute $1000 per month. At the end of the next 15 years, how much is her account worth?

To determine how much Diane's account will be worth at the end of 15 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial contribution)
r = annual interest rate (7.72% or 0.0772)
n = number of times that interest is compounded per year (monthly, so n = 12)
t = number of years

In this case, Diane contributes $1000 per month, so her principal investment amount (P) will increase by $1000 every month. Therefore, we need to calculate the future value of each monthly contribution and sum them up.

Let's break down the calculation into steps:

Step 1: Convert the annual interest rate to a monthly interest rate:
Monthly interest rate (r) = (1 + annual interest rate)^ (1/12) - 1
r = (1 + 0.0772)^ (1/12) - 1
r ≈ 0.00637 (approximately)

Step 2: Calculate the future value of each monthly contribution and sum them up.

For the first month:
A1 = P(1 + r)^nt
A1 = $1000(1 + 0.00637)^(12*1)
A1 ≈ $10,651.33

For the second month:
A2 = $1000(1 + 0.00637)^(12*2)
A2 ≈ $21,430.43

...

For the 15th year:
A15 = $1000(1 + 0.00637)^(12*15)
A15 ≈ $275,921.80

Step 3: Sum up all the future values of monthly contributions:
Account worth = A1 + A2 + A3 + ... + A15

Account worth ≈ $10,651.33 + $21,430.43 + ... + $275,921.80

By performing this calculation, we can determine the final value of Diane's account after 15 years.