(a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

How long will an account worth $\$ 2.2$ million provide $\$ 10,000$ at the end of each month, if money is worth $5.4 \%$ compounded monthly?

What is the future value if $\$ 8600$ is invested for $8$ years at $10 \%$ compounded semiannually?

Microsoft's stock price peaked at $6118 \%$ of its IPO price more than $13$ years after the IPO (Source: Bloomberg Financial Markets). Suppose that $\$ 10,000$ invested in Microsoft at its IPO price had been worth $\$ 600,000$ ( $6000 \%$ of the IPO price) after exactly $13$ years. What interest rate compounded annually does this represent?

(a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

A $\$ 2.4$ million state lottery pays $\$ 10,000$ at the beginning of each month for $20$ years. How much money must the state actually have on hand to set up the payments for this prize if money is worth $6.3 \%$ compounded monthly?

(a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

Sam deposits $\$ 500$ at the end of every $6$ months in an account that pays $8 \%$ compounded semiannually. How much will he have at the end of $8$ years?

Jadele, Inc., borrowed $\$ 12.8$ million at $7.2 \%$ compounded quarterly for $30$ years for construction of a new manufacturing facility. After making $42$ quarterly payments of $\$ 261,094.80$, it plans to refinance the existing loan for an amount that includes an additional $\$ 1.1$ million for expansion. Jadele can refinance at $6.6 \%$ compounded quarterly for $25$ years, with refinancing costs of $\$ 10,000$ added to the refinanced amount.

(a) Find the total amount refinanced.

(b) Find the new quarterly payment.

( c ) If Jadele decides to continue to pay $\$261,094.80$ per quarter on the refinanced amount, how many payments will it take to pay off the new loan?

Write the sixth term of the sequence whose $n$th term is

$$a_n=\frac{n(n-1)}{n+3}$$

Annuity What lump sum would have to be invested at $9 \%$ compounded semiannually to provide an annuity of $\$ 10,000$ at the end of each half-year for $10$ years?

(a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

A couple has determined that they need $\$ 1.5$ million to establish an annuity when they retire in $25$ years. How much money should they deposit at the end of each month in an investment plan that pays $10 \%$ compounded monthly, so that they will have the $\$ 1.5 \mathrm{mil}$ lion in $25$ years?

Time to reach a goal If $\$ 1200$ is deposited at the end of each quarter in an account that earns $7.2 \%$ compounded quarterly, how long will it be until the account is worth $\$ 60,000$ ?

(a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

Recent sales of some real estate and record profits make it possible for a manufacturer to set aside $\$ 800,000$ in a fund to be used for modernization and remodeling. How much can be withdrawn from this fund at the beginning of each half-year for the next $3$ years if the fund earns $7.7 \%$ compounded semiannually?

A young couple buying their first home borrow $\$ 85,000$ for $30$ years at $7.2 \%$ compounded monthly and make payments of $\$ 576.97$. After $3$ years, they are able to make a one-time payment of $\$ 2000$ along with their $36$th payment.

(a) Find the unpaid balance immediately after they pay the extra $\$ 2000$ and their $36$th payment.

(b) How many regular payments of $\$ 576.97$ will amortize the unpaid balance from part (a)? Give the answer to one decimal point.

( c ) How much will the remaining debt be after the number of full payment periods in (b) is made? How much extra must be included with last full payment to pay off the debt?

(d) How much will the couple pay over the life of the loan by paying the extra $\$ 2000$?

Write the first four terms and the 10 th term of the sequence whose $n$th term is

$$a_n=\frac{n-4}{n(n+2)}$$

A debt of $\$ 8000$ is to be amortized with $8$ equal semiannual payments. If the interest rate is $12 \%$ compounded semiannually, what is the size of each payment?