Algebra II / Mr. Hansen

Name: _________________________________

1.

A geometric series has first term 28 and common ratio –1.5. Compute the twentieth term and the twentieth partial sum. Use correct notation.

2.

An arithmetic series has first term –15 and common difference 2.1. Compute the thirtieth term and the thirtieth partial sum. Use correct notation.

3.

An arithmetic series has t₁ = 44 and t₁₀₀ = 88. Find t₂ and S₁₀₀.

4.

A series has the property that each term equals 91% of the term before. The first term is 815. Compute t₂ and t₃. What kind of series is this? Is it possible to add up infinitely many terms of this sequence? Why or why not? If possible, compute the infinite sum.

5.

Dilbert has a rich uncle (Uncle Bert) who wishes to award him $2,000 on Dilbert’s 40th birthday. On May 1, Dilbert will be exactly 23 years and 10 months old, and Dilbert was born on July 1 (the start of a calendar quarter). Uncle Bert is fairly old and may not live to see Dilbert’s 40th birthday. Therefore, on May 1, 1999, Uncle Bert will deposit a certain amount (call it x) in a bank account that pays a guaranteed 4.1% interest rate, compounded quarterly. The bank computes interest in whole months only (for example, an amount on deposit for 2 months would earn an interest amount of (2/12) ´ 0.041 ´ (amount on deposit at beginning of period) since 2/12 is the fraction of the year that 2 months represents, and the annual interest rate of 4.1% can be written as 0.041.

Since Uncle Bert has studied geometric sequences, he realizes that the value of the $2,000 if left in the savings account after his 40th birthday would be a term in a geometric sequence, with independent variable p being the number of quarters the money has been left. The value of the $2,000 before his 40th birthday, therefore, will be given by the same sequence, but with the appropriate negative value substituted for p. How much should Dilbert ask his uncle to invest? (In other words, what is x such that after 16 years and 2 months, x will grow to $2,000?) Businesspeople call this amount the "present value" (abbreviated PV) of the $2,000.

There are many other ways to work this problem as well. Show all your work. Neatness counts. Round all intermediate results to the nearest penny. Make your writeup clear and sequential, so that somebody (e.g., your teacher) can figure out what your thought process was. You should write a few complete sentences here and there to make the presentation understandable. You may need to do a scratch version and then write a better version to submit. If you have a spreadsheet or a financial calculator, you can use a PV function to check your work, but be aware that the result probably won’t match precisely.