Chess legend If in the previous problem the king granted the inventor's wish for half a chessboard, how many grains would the inventor receive?

Chess legend Legend has it that when the king of Persia wanted to reward the inventor of chess with whatever he desired, the inventor asked for one grain of wheat on the first square of the chessboard, with the number of grains doubled on each square thereafter for the remaining $63$ squares. Find the number of grains on the $64$th square. (The sum of all the grains of wheat on all the squares would cover Alaska more than $3$ inches deep with wheat!)

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

An insurance settlement of $\$ 1.5$ million must replace Trixie Eden's income for the next $40$ years. What income will this settlement provide at the end of each month if it is invested in an annuity that earns $8.4 \%$ compounded monthly?

Find the annual percentage yield for an investment at

(a) $7.3 \%$ compounded monthly

(b) $6 \%$ compounded continuously.

A developer wants to buy a certain parcel of land. The developer believes she can afford payments of $\$ 44,000$ each half-year for the next $7$ years. How much can she borrow and hold to this budget at each of the following interest rates?

(a) $8.9 \%$ compounded semiannually

(b) $7.3 \%$ compounded semiannually

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

Parents agree to invest $\$ 500$ (at $10 \%$ compounded semiannually) for their son on the December $31$ or June $30$ following each semester that he makes the dean's list during his $4$ years in college. If he makes the dean's list in each of the $8$ semesters, how much money will his parents have to give him when he graduates?

Bill Casler bought a $\$ 2000$, $9$-month certificate of deposit ($\mathrm{CD}$) that would earn $8 \%$ annual simple interest. Three months before the $\mathrm{CD}$ was due to mature, Bill needed his $\mathrm{CD}$ money, so a friend agreed to lend him money and receive the value of the $\mathrm{CD}$ when it matured.

(a) What is the value of the $\mathrm{CD}$ when it matures?

(b) If their agreement allowed the friend to earn a $10 \%$ annual simple interest return on his loan to Bill, how much did Bill receive from his friend?

A couple purchasing a home budget $\$ 1800$ per month for their loan payment. If they have $\$ 20,000$ available for a down payment and are considering a $25$-year loan, how much can they spend on the home at each of the following rates?

(a) $6.9 \%$ compounded monthly

(b) $7.5 \%$ compounded monthly

A woman makes $\$ 3000$ contributions at the end of each half-year to a retirement account for a period of $8$ years. For the next $20$ years, she makes no additional contributions and no withdrawals.

(a) If the account earns $7.5\%$ compounded semiannually, find the value of the account after the $28$ years.

(b) If this account is used to set up an annuity that pays her an amount at the beginning of each $6$-month period for the next $20$ years, how much will each payment be?

Seana wants to save $\$ 20,000$ toward the purchase of a new car. She can contribute $\$ 400$ at the beginning of each month into an account that earns $4.8 \%$ compounded monthly. How long until she reaches her goal?

How much more interest will be earned if $\$ 5000$ is invested for $6$ years at $7 \%$ compounded continuously, instead of at $7 \%$ compounded quarterly?

Comparing yields Find the annual percentage yield equivalent to a nominal rate of $7.2 \%$ (a) compounded quarterly and (b) compounded continuously.

A couple inherits $\$ 89,000$. How much can this generate at the beginning of each month over the next $5$ years, if money is worth $6.3 \%$ compounded monthly?

A year-end bonus of $\$ 25,000$ will generate how much money at the beginning of each month for the next year, if it can be invested at $6.48 \%$ compounded monthly?