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1. According to the National Education Association, the average classroom teacher in the US earned $43,837 in annual salary for the 1999-2000 school year.
a. If the teachers receive an average salary increase of $1096, write out the first 6 terms of the sequence formed by the average salaries starting with the 1999-2000 school year. Explain how you got your answer. (1 point)
b. Write the general form for the sequence. (1 point)
c. Write the recursive formula for an. (1 point)
2. In 1965, Gordon Moore, the cofounder of Intel, predicted that the number of transistors that could be designed into an integrated circuit would double every two years. This result is known as Moore’s Law.
a. Complete the following table, showing the number of transisters per circuit for the indicated years.
(1 point)
b. Express this sequence using a recursive formula in which we can express any term an in terms of the term an-2 (the term 2 years prior). [Hint: Remember that n represents the number of years since 1970, since n = 2 represents the year 1972. ] (1 point)
c. According to Intel, the Pentium 4 processor circuit, released in the year 2000, is designed using 42,000,000 transistors. According to your calculations, is this circuit consistent with Moore’s Law? Explain your answer. (1 point)
3. Expand the following summation, then evaluate. In your explanation, describe the steps involved in arriving at your answer. (5 points)
4. The sequence formed by the Lucas numbers is as follows: . Using proper terminology as you learned in this unit, compare and contrast the Lucas sequence with the famous Fibonacci sequence by naming at least one similar property and one contrasting property. (4 points)
Part II. Case Study The Mystery of the Missing Coulomb
This week Patty Madeye is going to be investigating the theft of a rare Orange Tiger Coulomb (shown at the right), which is owned by Madame Levare, who lives in West Floflux.
Since the jewels are quite valuable, Madam Levare stores them in the vault at the jeweler’s store, West FloFlux GemStone in downtown West FloFlux. Only certain lockboxes in the vault were touched – it seems that the thief knew exactly what he was looking for.
Task #1 – Patty’s first task is to determine the value of the jewels. She talks to the jeweler who created them and he estimates the value of the jewels in 1985 (when they were purchased) at $65,000. The value is thought to increase (appreciate) by $1500 per year. If this is true, how much would the jewels be worth in 2010? Explain how you arrived at your answer. (4 points)
Task #2 - Patty talks to the jeweler and discovers that he remembers the 4-digit combination to the main vault in the store by writing it down in summation form. Here’s what he wrote:
The combination is written in summation form, but some of the notation is cut off from where the paper is ripped. You’ll need to figure out the full equation so that Patty can get into the vault to investigate. (4 points)
Task #3 - Patty notices a pattern in the numbers of the lockboxes that were touched during the robbery and says that she thinks that it’s a mathematical sequence. The sequence is { 8, 15, 22, 29, …}. Determine whether this is a sequence (as far as you can tell) and what type (arithmetic or geometric) it is. Justify your answer by stating the general term for the sequence. Assuming Patty is correct, can you identify two other lockboxes that might have been emptied using this sequence? (4 points)
Task #4 (8 points) – Patty asks you to find out more information about the Fibonacci sequence as background for this week’s episode. Do some research on the Fibonacci numbers by consulting the Kaplan Library or the internet. Find two facts or interesting properties about this fascinating topic and write a 1 page essay describing what you have found. Possible approaches include:
- The origin of the Fibonacci sequence?
- What is the connection between the Fibonacci sequence and the golden ratio?
- Does the “golden string” ever repeat?
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