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 simple interest earned on an investment

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

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

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 $\$ 45,000$?

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?

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.

Kodicom Inc. has $15$-year bonds with a $\$ 5000$ maturity value and a quoted coupon rate of $12 \%$ paid semiannually. The current yield is $10 \%$ compounded semiannually.

(a) Compute the price of these bonds.

(b) Suppose that with $12$ years remaining until maturity, the yield rate drops to $8 \%$ compounded semiannually. Find the new price of these bonds.

Amortization schedule Complete the amortization schedule for the next two payments of a loan for $30$ years at $7.5 \%$ compounded monthly, with monthly payments of $\$ 699.22$.

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?

Find the sum of the first $56$ terms of the arithmetic sequence with first term $6$ and $56$th term $226$.

(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.

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

Develop an amortization schedule for a $4$-year car loan if $\$ 16,700$ is borrowed at $8.2 \%$ compounded monthly.