Objective: 1. After completing activity 1, mod With 90% accuracy 3. -Identify sequences as arithmetic, geometric, or neither -Write recursive formulas for arithmetic and geometric sequences -Write explicit formulas for arithmetic and geometric sequences -Determine the number of terms in a finite arithmetic or geometric sequence

Arithmetic Sequence: An arithmetic (linear) sequence is a sequence of numbers in which each new term ( )is calculated by adding a constant vale(d) to the previous term. For example: 1,2,3,4,5,6,… The value of d is 1. Find the constant value that is added to get the following sequences & write out the next 3 terms. 1. 2,6,10,14,18,22,…

Simple test to check if a pattern is an arithmetic sequence: Check that the difference between consecutive terms is constant. For example, in the sequence: 1,2,3,4,5,6,… the constant is one because 6-5=5-4=4-3=3-2=2-1=1 In other words, since the difference is constantly 1, then it is an arithmetic sequence

Find the constant value that is added to get the following sequences & write out the next 3 terms 2,6,10,14,18,22,… (you did this one already) -5,-3,-1,1,3,… 1,4,7,10,13,16,… -1,10,21,32,43,54,… 3,0,3,-6,-9,-12,…

The recursive formula for an arithmetic sequence For example, the recursive formula for the arithmetic sequence 1,2,3,4,5,6… is

Write the recursive formula for each sequence: 2,6,10,14,18,22,… -5,-3,-1,1,3,… 1,4,7,10,13,16,… -1,10,21,32,43,54,… 3,0,3,-6,-9,-12,…

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Explicit formula The explicit formula for a sequence defines any term based on its term number (n):

Write the explicit formula for each sequence 2,6,10,14,18,22,… -5,-3,-1,1,3,… 1,4,7,10,13,16,… -1,10,21,32,43,54,… 3,0,3,-6,-9,-12,…

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Use your explicit formulas to answer the questions: (show your work) 1. What is the third term in the pattern: 2,6,10,14,18,22,… 2.What is the 20 th term? 3.What is the 35 th term?

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Geometric Sequence: A geometric sequence- every term after the first is formed by multiplying the preceding term by a constant value called the common ratio (or r) For example: 2,10,50,250,1250 ◦ The value of r is 5 because :

Simple test to check if a sequence is a geometric sequence: When you divide a term by a previous term you must arrive at equal common ratios.

Determine the common factor for the following geometric sequence: 5,10,20,40,80,… 7,28,112,448,… 2,6,18,54,…

Answer: 2 ½ 4 3

The recursive formula for a geometric sequence Write the recursive formula for each geometric sequence : 5,10,20,40,80,… 7,28,112,448,… 2,6,18,54,…

Write the recursive formula for each geometric sequence 5,10,20,40,80,… 7,28,112,448,… 2,6,18,54,…

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The explicit formula for a geometric series is: Write the explicit formula for each geometric sequence : 5,10,20,40,80,… 7,28,112,448,… 2,6,18,54,…

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Is the following sequence arithmetic or geometric?: -3,30,-300,3000,…. Write a recursive & explicit formula for it. Use the explicit formula to find the 8 th term.

Answer: Geometric

Warm-up 3/8/10 State whether the sequence is arithmetic, geometric, or neither. Use your notes.

Answers: Arithmetic Neither Geometric

Warm-Up 3/9/10 (head your paper) Consider the following arithmetic sequences: 0, 6, 12, 18,…120 1, 9, 17, 25,…97 15, 12, 9, 6,…-21 What is the common difference for each? How many terms are in the sequence?

Class work: Remember to head your notebook with page # & today’s date. Old green Alg. II Book page 476 #1-9 under Exercises & Applications also do #13 & 14. Be Ready for a Quiz on it! Your regular book: Page 277 Assignment #1.1, 1.3, 1.4, 1.9

Class work: 3/9/10 Old green Alg. II book page 479 #35-37 Read the problem carefully Your regular book Page 277 Assignment #1.5 a-c

Homework: Page a-f. Page 277 Warm-up #1-2

Homework Review, Check your work: Page 274 Discussion a. 1. arithmetic 2. geometric 3. neither 4. geometric 5. neither 6. Fibonacci

Discussion b.

Objective: 1.After completing activity 2, mod With 90% accuracy 3.-Identify sequences as arithmetic, geometric, or neither Write explicit formulas for arithmetic and geometric sequences Determine the number of terms in a finite arithmetic sequence Write formulas for finite arithmetic series

Activity 2 Notes Finite Series- the sum ( )of the terms of a finite sequence. ◦ For example: a finite series with n terms is: ◦ Arithmetic Series: the sum of the terms of an arithmetic sequence. For example:

Find the sum of the first 100 natural numbers:

Activity 2 Notes Class work: page 281 exploration Parts a-c only

Answers to Exploration:

Conclusion Question:

Formula:

Formula 2: The sum of the terms of a finite arithmetic sequence with n terms & a common difference d can also be found by using the formula: Please notice that this formula involves the common difference d.

Warm-up: 3/11/10 Consider the sequence: 7,11,15,…59 Find the sum of all the terms Answer: 462

Homework: 3/10/10 Warm –up page #1-3 Check your HW: 1. 2,001,000 2a. )3 2b.) 4 2c.) 113 2d. ) 25,561 3a.) 780 3b.) 1197

Class work: 3/11/10 Assignment page # 2.1, 2.3, 2.4, 2.5, 2.6, 2.7

Answers to Assignment: 2.1) sum of first n even numbers: n(n+1) 2.3) No because the sum of each pair is not a constant 2.4a.) the monthly payments can be considered to be an arithmetic sequence where the first term is $ and the common difference is $ B) Yes. The payments form an arithmetic sequence, their sum forms a series.

Answers to Assignment: 2.4c) The lessee pays 2.4d) The difference of $ may be the cost to purchase the car at the end of the lease.

Answers to Assignment: 2.5) 2.6a) -2,3,8 2.6b) c) 55,575

Answers to Assignment: 2.7a) b) c) and 10.75

Objective 1.After completing activity 3, mod With 90% accuracy 3.Identify sequences as arithmetic, geometric, or neither Write explicit formulas for arithmetic and geometric sequences Determine the number of terms in a finite geometric sequence Write formulas for finite geometric series

Geometric Series: Geometric Series- the sum of the terms of a geometric sequence. For example: 2,6,18,54,162

Explore: Head your notebook with today’s date, page # & title. With your partner, try the exploration on page parts a-g

Formula: The sum of a finite geometric series with n terms and a common ratio r: Use the formula with the geometric sequence: 2,6,18,54,162 to find the sum of all 5 terms.

Warm-up page #1-3 Assignment: # Skip part c for #3.3 Quiz Activity 2& 3 on Tuesday: Arithmetic & Geometric Series. You must know Activity 1 to pass the quiz. 

warm-up

Objective: 1.After completing activity 4, mod With 90% accuracy 3. Write explicit formulas for arithmetic and geometric sequences Interpret the limit of an infinite sequence Determine the sum of the terms of an infinite geometric sequence in which the common ratio r is between –1 and 1 Compare sequences that do and do not approach limits