HONORS FST - Dearborn Public Schools



HONORS FST CHAPTER 8

SECTION 8-1

Homework: LM 8-1

New Terms:

1. Term:

2. Position:

3. Explicit Formula:

4. Recursive Formula:

5. Arithmetic Sequence:

6. Geometric Sequence:

Definitions:

1. Sequence

Theorems:

1. Let n be a positive integer and a1 and d be constants. The formulas

a1

an= a1 + (n –1)d and

an = an-1 + d, n > 1

generate the terms of the arithmetic sequence with first term a1and constant difference d.

2. Let n be a positive integer and g1 and r be constants. The formulas

g1

gn= g1r n –1 and

gn = rgn-1, n > 1

generate the terms of the geometric sequence with first term g1 and constant ratio r.

Teacher Examples:

1. a. What is the 9th term of the sequence 2, 4, 6, 8,…?

b. Did you use an explicit formula or a recursive formula to get the 9th term in part a.

2. Give a recursive formula for the increasing sequence of positive odd numbers.

3. Which term is 344 in the arithmetic sequence 8, 15, 22, 29,…?

4. Suppose a model train collection valued now at $2000 increases 5% in value each year.

a. Give its value next year.

b. Give the value of the collection n years from now.

c. What will its value be 10 years from now?

SECTION 8-2

Homework: LM 8-2

New Terms:

7. divergent/diverge:

8. convergent:

9. harmonic sequence:

10. alternating harmonic sequence:

List all of the Limit Properties (pg 497, 499):

1. The limit of the harmonic sequence is 0.

Given: lim an and lim bn exist and c is a constant, then:

n([pic] n([pic]

2. The limit of a constant sequence is that constant

3.

4. The limit of a sum or product is the sum or product of the individual limits.

5. The limit of a constant times the terms of a sequence equals the product of the constant and the limit of the sequence.

Given: lim an and lim bn exist and lim bn [pic]0, then:

n([pic] n([pic] n([pic]

6. The limit of a reciprocal is the reciprocal of the limit.

7. The limit of a quotient is the quotient of the individual limits.

Teacher Examples:

1. Is the sequence defined by Sn = ___n2__ convergent or divergent? If

2n + 100

it is convergent, give its limit.

2. Find lim (-1)n.

n([pic] n2

3. Find lim 6n + 4.

n([pic] 5-3n

SECTION 8-3 Homework: LM 8-3

New Terms:

11. series:

12. infinite series:

13. finite series:

14. finite/infinite arithmetic series:

Theorems:

1. The sum Sn = a1 + … + an of an arithmetic series with first term a1 and constant difference d is given by

Sn = [pic](a1 + an) or Sn = [pic](2a1 + (n-1)d)

Teacher Examples:

1. Evaluate:

a. 8

( (n2 – 1)

n=1

b. 8

( (n2) – 1

n=1

2. A packer had to fill 100 boxes identically with machine tools. The shipper filled the first box in 13 minuets, but got faster by the same amount each time as time went on. If he filled the last box in 8 minutes, what was the total time that it took to fill the 100 boxes?

3. A new business decides to rank its 9 employees by how well they work and pay them amounts that are in arithmetic sequence with a constant difference of $500 a year. If the total amount paid the employees is to be $250,000, what will the employees make per year?

SECTION 8-4

Homework: LM 8-4

New Terms:

15. geometric series:

16. nth partial sum:

Theorems:

1. The sum Sn = g1 + g2 + … + gn of the finite geometric series with first term g1 and constant ratio r [pic]1 is given by Sn = [pic].

Teacher Examples:

1. Evaluate:

6

( 10(0.75)i - 1

i=1

2. In a set of 10 Russian nesting dolls, each doll is 5/6 the height of the taller one. If the height of the first doll is 15 cm, what is the total height of the dolls?

3. Suppose you have two children who marry and each of them has two children. Each of these offspring has two children, and so on. If all of these progeny marry but none marry each other, and all have two children, in how many generations will you have a thousand descendants? Count your children as Generation 1.

SECTION 8-5

Homework: LM 8-5

New Terms:

17. convergent:

18. divergent:

Definitions:

1. The sum S[pic] of the infinite series is the limit of the sequence of partial sums Sn of the series, provided the limit exists and is finite. In symbols,

If Sn = [pic]

Then S[pic]= [pic] = [pic]Sn = [pic][pic]

Theorems:

1. Consider the infinite geometric series

g1 + g1r + g1r2 + …+ g1rn-1 +…, with g1 [pic]0.

a. If abs(r) < 1, then the series converges and S[pic] = [pic].

b. If abs(r) [pic] 1, the series diverges.

Teacher Examples:

1. Consider the infinite geometric series 1+.99 +.992,+ .993+…+ .99n-I +..

a. Find the first five partial sums.

b. Does the sequence of partial sums seem to converge? If so, what is the value of the series?

2. How much distance will the Superball in Question 17 of Lesson 8-4 traverse before it comes to rest?

3. The infinite series 1 + 2/1! + 4/2! + 8/3! + … + 2n/n! + … converges. Use a calculator or computer to approximate the limit to six decimal places.

SECTION 8-6

Homework: LM 8-6

New Terms:

19. Combination

20. nCr

(find this key on you calculator)

Questions:

1. How are Combinations and Permutations different? Explain.

2. How are their formulas related? Explain.

Theorems:

1. Formula for nCr

Teacher Examples:

1. Twenty distinct points are chosen on a circle.

a. How many segments are there with these points as endpoints?

b. How many triangles are there with these points as vertices?

c. How many quadrilaterals are there with these points as vertices?

2. Which is the permutation problem, which is the combination problem?

a. You have six colors to choose from and you wish to choose three for a flag. How many choices of colors are possible?

b. A flag with strips of three different colors can use any one of six colors. How many flags are possible?

3. At a restaurant you can order pizza with any of 9 different toppings. How many different pizzas are possible with exactly 3 of those toppings?

4. Five students from your class are randomly picked to be interviewed by the local newspaper. If your class contains 20 students, what is the probability that neither you nor your best friend in the class will be picked?

SECTION 8-7

Homework: LM 8-7

New Terms:

21. Pascal’s Triangle

Definitions:

1. Let n and r be nonnegative integers with r [pic] n. The (r + 1)st term in row n of Pascal’s Triangle is nCr.

Teacher Examples:

5. Use the formula for nCr to find the first four terms in row 9 of Pascal’s Triangle.

6. Give a polynomial formula for the third term in row “n” of Pascal’s Triangle.

7. Here are the first 6 terms of row 11 of Pascal’s Triangle: 1, 11, 55, 165, 330, 462. Use them to construct row 12.

SECTION 8-8

Homework: LM 8-8

New Terms:

22. binomial coefficients

The Binomial Theorem:

1. For any nonnegative integer n,

(x + y)n = nCoxn + nC1xn-1y + nC2xn-2y2 + nCkxn-kyk + …+ nCnyn

=[pic]nCkxn-kyk.

Teacher Examples:

1. Give the coefficient of a9b2 in (a + b)11.

2. Expand (4x +5y)5

3. Expand (4x2 +5y)5

4. A coin is flipped 6 times. How many of the possible outcomes have at least 3 heads?

SECTION 8-9

Homework: LM 8-9

New Terms:

23. binomial experiment

Theorems:

1. Binomial Probability Theorem:

Suppose that in a binomial experiment with n trials the probability of success is p in each trial, and the probability of failure is q, where q = 1 – p. Then

Teacher Examples:

1. The probability of getting a sum of 7 is a toss of two fair die is known to be 1/6.

a. What is the probability of getting exactly two 7s in 5 tosses?

b. What is the probability of getting at least two 7s in 5 tosses?

2. What is the probability of getting at least 8 out of 10 questions on a test correct, if you feel you have an 80% chance of answering each individual question correctly?

3. Determine the probability distribution from the binomial experiment of Additional Example 1.

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