8. Mary would like to save $10 000 at the end of 5 years for a future down payment on a car. How much should she deposit at the end of each week in a savings account that pays 12%/a, compounded monthly, to meet her goal?
a. What type of annuity is this?
c. Write the series that represents the amount of annuity.
d. Determine the weekly deposit without technology.
a. This is an ordinary annuity since the deposits are made at the end of each week.
c. To find the series that represents the amount of the annuity, we'll use the formula for the future value of an ordinary annuity:
FV = PMT * [(1+r)^n - 1] / r
Where:
FV = Future value of the annuity (which is $10,000 in this case)
PMT = Weekly deposit
r = Interest rate per period (which is 12% per annum divided by 12 months = 1% per month = 0.01)
n = Number of periods (which is 5 years multiplied by 12 months = 60 months)
Substituting the values, we get:
10,000 = PMT * [(1+0.01)^60 - 1] / 0.01
Simplifying:
10,000 = PMT * [1.01046082831 - 1] / 0.01
10,000 = PMT * 0.01046082831 / 0.01
10,000 = PMT * 1.046082831
Dividing both sides by 1.046082831:
10,000 / 1.046082831 = PMT
9,553.89 ≈ PMT
So, the weekly deposit should be approximately $9,553.89.
d. To determine the weekly deposit without using technology, you can use a financial calculator or an online annuity calculator to find the future value of the annuity. Then use trial and error to find the PMT (weekly deposit) that results in a future value closest to $10,000. Start with an initial guess for the PMT and adjust it up or down until you find the correct value.