The problem is complex financial problems that require several skills, perhaps some from previous sections.

A small business owner contributes $\$ 3000$ at the end of each quarter to a retirement account that earns $8 \%$ compounded quarterly.

(a) How long will it be until the account is worth $\$ 150,000$?

(b) Suppose when the account reaches $\$ 150,000$, the business owner increases the contributions to $\$ 5000$ at the end of each quarter. What will the total value of the account be after $15$ more years?

The problem is complex financial problems that require several skills, perhaps some from previous sections.

Suppose a young couple deposits $\$ 1000$ at the end of each quarter in an account that earns $7.6\%$ compounded quarterly for a period of $8$ years. After the $8$ years, they start a family and find that they can contribute only $\$ 200$ per quarter. If they leave the money from the first $8$ years in the account and continue to contribute $\$ 200$ at the end of each quarter for the next $18 \frac{1}{2}$ years, how much will they have in the account (to help with their child's college expenses)?

Identify which of the following formulas applies to each situation in the problem.

$$\begin{gathered} I=\operatorname{Prt} \quad S=P(1+i)^n \quad S=P e^{n t} \\ S=R\left[\frac{(1+i)^n-1}{i}\right] \quad A_n=R\left[\frac{1-(1+i)^{-n}}{i}\right] \end{gathered}$$

The future value of a series of payments, with interest compounded when the payments are made

How long does it take an account containing $\$ 8000$ to be worth $\$ 15,000$ if the money is invested at $9 \%$ compounded monthly?

Find the sum of the first $80$ terms of the arithmetic sequence with first term $12$ and common difference $-3$.

A friend comments to you that there was a beautiful, thin sliver of Moon visible in the early morning just before sunrise. Which phase of the Moon would this be, and in what direction would you look to see the Moon (in the southern sky, on the eastern horizon , on the western horizon, high in the sky, etc.)?

Use a spreadsheet or financial program on a calculator or computer to complete the problems.

Develop an amortization schedule for a $10$-year mortgage loan of $\$ 80,000$ at $7.2 \%$ compounded monthly.

The problem is complex financial problems that require several skills, perhaps some from previous sections.

Suppose a recent college graduate's first job allows her to deposit $\$ 100$ at the end of each month in a savings plan that earns $9 \%$ compounded monthly. This savings plan continues for 8 years before new obligations make it impossible to continue. If the accrued amount remains in the plan for the next $15$ years without deposits or withdrawals, how much money will be in the account $23$ years after the plan began?

A couple needs $\$ 45,000$ as a down payment for a home. If they invest the $\$ 30,000$ they have at $8 \%$ compounded quarterly, how long will it take the money to grow to $\$ 45,000$?

What causes the phases of the moon?

Find the sum of the first $70$ terms of the arithmetic sequence with first term $10$ and common difference $\frac{1}{2}$.

Loan refinancing A corporation borrowed $\$ 750,000$ for $15$ years at $5.4 \%$ compounded semiannually. After making $17$ payments of $\$ 36,795.60$, it refinances at $4.8 \%$ compounded semiannually for $5$ years, with refinancing costs of $\$ 2200$ added to the new loan.

(a) Find the amount of the new loan.

(b) Find the new semiannual payment.

( c ) Find the amount saved by refinancing.

A property owner has several rental units and wants to build more. How much of each month's rental income should be deposited at the beginning of each month in an account that earns $6.6 \%$ compounded monthly, if the goal is to have $\$ 100,000$ at the end of $4$ years?

(a) The purchase of Alaska cost the United States $\$ 7$ million in $1869$. If this money had been placed in a savings account paying $6 \%$ compounded annually, how much money would have been available from this investment in $2015$?

(b) If the $7 million earned 7% compounded annually since$1869$, how much would have been available in$2015$?

( c ) Do you think either amount would have purchased Alaska in$2015$? Explain in light of the value of Alaska's resources or perhaps the price per acre of land.