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

(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.

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 $\$ 699.22$.

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

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?

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.

Cash value A woman paid $\$ 90,000$ down for a house and agreed to pay $18$ quarterly payments of $\$ 4500$ each. If money is worth $4 \%$ compounded quarterly, how much would the house have cost if she had paid cash?

Find the sum of the first $38$ terms of the arithmetic sequence with first term $2$ and $38$th term $113$.

A 10-year Emmaco Corporate bond has a par value of $\$ 10,000$ with coupons at $7.8 \%$ paid semiannually.

(a) If this bond is bought to yield $10 \%$ compounded semiannually, find its price.

(b) Suppose that after this bond has been held for $2$ years, the desired yield is $8 \%$ compounded semiannually. Find the selling price.

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

Clark and Lana take a $30$-year home mortgage of $\$ 121,000$ at $7.8 \%$ compounded monthly. They make their regular monthly payments for $5$ years, then decide to pay $\$ 1000$ per month.

(a) Find their regular monthly payment.

(b) Find the unpaid balance when they begin paying the $\$ 1000$.

( c ) How many payments of $\$ 1000$ will it take to pay off the loan? Give the answer correct to one decimal place.

(d) Use your answer to part ( c ) to find how much interest they save by paying the loan this way.

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

A sinking fund is established by a working couple so that they will have $\$ 60,000$ to pay for part of their daughter's education when she enters college. If they make deposits at the end of each $3$-month period for $10$ years and if interest is paid at $12 \%$ compounded quarterly, what size deposits must they make?

(a) A $40$-year-old man has $\$ 432,860$ in an IRA account. He decides to make no additional contributions to the account but expects it to grow at $7.5 \%$ compounded annually. How much does he expect to have in the account when he retires at age $62$?

(b) How much more money would the man have if his investments earned $8.5 \%$ compounded annually?

A debt of $\$ 280,000$ is amortized with $40$ equal semiannual payments of $\$ 14,357.78$. If interest is $8.2 \%$ compounded semiannually, find the unpaid balance of the debt after $25$ payments have been made.

Find the $73$rd term of the arithmetic sequence with first term $20$ and $10$th term $47$.