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Mathematical Patterns. Arithmetic Sequences. Arithmetic ... · PDF file Arithmetic Sequences. Arithmetic Series. To identify mathematical patterns found n a sequence. To use a formula

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  • Mathematical Patterns. Arithmetic Sequences.

    Arithmetic Series.

    To identify mathematical patterns found n a sequence. To use a formula to find the nth term of a sequence. To define, identify, and apply arithmetic sequences. To define arithmetic series and find their sums.

    Vocabulary Sequence, term of a sequence, explicit formula, recursive formula. Arithmetic sequence, common difference. Arithmetic series,

  • What is a sequence? A sequence: is an ordered list of numbers. Each number in a sequence is a term of a sequence.

    1st term

    11321 , , ,.... , ,  nnn aaaaaa 2nd term

    n-1 term

    nth term

    n+1 term

    An explicit formula describes the nth term of a sequence using the number n. It is used to represent terms or find a term.

    For example: 2,4,6,8,10,. . . . The nth term is twice the value of n. nan 2

    A recursive formula: a formula that describes the nth term of a sequence by referring to preceding terms.

    For example: 133,130,127,124, . . .

    condition initial 1331 a 1nfor ,31  nn aa

  • Problem 1: Generating a Sequence using on Explicit Formula

    Answer: 2n3an 

      1213a1 

      4223a2  1,4,7,10,13,16,19,22,25,28.

    Your turn ?e sequenceterm in ththe a. What is nan 12312 

    Answer: 147

    nce?this seque terms of

    the first. What arenula alicit form has an A sequence n

    10

    23exp 

  • Problem 2: The number of blocks in two-dimensional pyramid is a sequence that follows a recursive formula. What is a recursive definition for the sequence?

    Count the number of blocks in each pyramid.

    Subtract consecutive terms to find out what happens from one term to the next.

    Use n to express the relationship between successive terms.

    1,3,6,10, 15,21

    213aa 12 

    336aa 23 

    4610aa 34  naa 1nn  

    To write a recursive definition, state the initial condition and the recursive formula.

    na1a 1n1  na and

    Answer

  • Problem 3: Using G.C. Using Formulas to find terms of a sequence.

    Pierre began the year with an unpaid of $300 on his credit card. Because he had not read the credit card agreement, he did not realize that the company charged 1.8% interest each month on his unpaid balance, in addition to a $29 penalty in any month he might fail to make a minimum payment. Pierre ignored his credit card bill for 4 consecutives months before finally deciding to pay off the balance. What did he owe after 4 months of no payment?

    Step 1: Write a recursive definition

    Step 2: Use a calculator(press MODE key)

    month. 1 afterbalance the represents a that soa use(300a :condition Initial 10 0 

    Answer

    After 4 months, Pierre owns 441.36

    1n a :formula ecursive n   29a018.1R 1n

  • Take a note: An arithmetic sequence is a sequence where the difference between any two consecutives terms is a constant. This difference is called common difference.

    A recursive definition for this sequence has two parts:

    formula recursive 1nfor a

    condition initial

    n 

     ,da

    aa

    1n

    1

    An explicit definition for this sequence is a single formula:

      1nfor an  ,d1na

    Examples of Arithmetic sequences:

    3,6,9,12,15,…

    1,4,9,16,25,…

    Yes it is A.S. and the difference is 3

    No, there is not a common difference

  • Problem 4: Analyzing Arithmetic Sequences.

    A. What is the 100th term of the arithmetic sequence that begins 6,11,…. ? B. What are the second and third terms of the arithmetic sequence 100, ___ ,____ , 82, ….?

    56-11 is difference common theand  ,6a1  d1naan 

     511006a100  501a100 

    Use the explicit formula Substitute and simplify

    The 100th term is 501

    82 and100 between sndifference common are 3 There .82athe and 100a 41 

    82=100+3d

    -18=3d

    -6=d

    The common difference is -6 and the terms are 100,94,88,82,…

  • Take a note: the arithmetic mean, or average of two Numbers x and y is . The arithmetic sequence, the middle term of any three consecutives terms is the arithmetic mean of the other two terms.

    2

    yx 

    Problem 5: Using Arithmetic Mean

    What is the missing term of the arithmetic sequence ….,15,___ , 59, …?

    Answer:

    37is termmissing the 37 2

    5915 

  • Problem 6: Using an Explicit Formula for an Arithmetic Sequence

    The numbers of seats in the first 13 rows in a section of an arena form an arithmetic sequence. Rows 1 and 2 are shown in the diagram below. How many seats are in row 13?

    Step 1: Find the common difference 21416aad 12 

    Step 2: Write an explicit formula for

     d1naan 

     211314a13  38 

    Row 2: 16 seats

    Row 1: 14 seats

  • Finite Sequence: 6, 9, 12, 15, 18

    Finite Series: 6+ 9+ 12+ 15+ 18 (The sum is 60)

    Infinite Sequence: 3, 7, 11, 15,….

    Infinite Series: 3 + 7 + 11 + 15 +

    An arithmetic series is a series whose terms form an arithmetic sequence . When a series has a finite number of terms, you can use a formula involving the first and last term to evaluate the sum.

    What is a series? Take a note: A SERIES is an indicated sum of the terms of a sequence.

  • ARITHMETIC SERIES

    The sum of n terms of an arithmetic series

     n1n aa 2

    n S 

  • Problem 1: Finding the sum of a Finite Arithmetic Series

    What is the sum of the even integers from 2 to 100?

    The series 2 + 4 + 6 + 8 +……+ 100 is arithmetic with the first term 2, last 100 and a common difference 2 and the sum is

     n1n aa 2

    n S 

     

    2550

    1002 2

    50 S50

    

    Your turn: What is the sum of the finite arithmetic series 4 + 9 + 14 + 19…. +99?

      1030994 2

    20 S20 

    Answer

    5d 4a1 

    n20

    1n599

    1n5

    5)1n(4

    d)1n(aa 1n

    

    

    

    

    Answer

    2d  100a2a n1

    d)1n(aa 1n 

    2)1n(2100 

    n2100 

    n50 

     n1n aa 2

    n S 

  • Problem 2: Using the sum of a finite arithmetic series A company pays a $10,000 bonus to salespeople at the end of the first 50 wks. if they make 10 sales in their first week, and then improve their sales numbers by two each week thereafter. One salesperson qualified for the bonus with the minimum possible number of sales. How many sales did the salesperson make in the week 50? In all 50 wks.?

    2d  10a1

    d)1n(a1 na

      10898102  1-5010

     n1n aa 2

    n S 

        29501182510810  2

    50

    The salesperson made 108 sales 1 wk. 50 and 2950 sales in all 50 wks.

    Use the explicit formula to find the sales in wk. 50

    Answer

  • You can use the Greek capital letter sigma to indicate a sum. You use limits to indicate the least and greatest value of n in the series. For example you can write the series as

    108642 

     

    5

    1n

    n2 108642 

    Read “the sum of 2n as n goes from 1 to 5”

    Take a note: To write a series in summation notation you need an explicit formula for the nth term and the lower and upper limits.

    Problem 3: Writing a series in Summation Notation. What is summation notation for the series 7 + 11 + 15 + . . . + 203 + 207 ?

     dnaa

    da

    n 1

    47

    1

    1

    

    

    n51

    3n4207

    3n4an

    

    

    (4𝑛 + 3)

    51

    𝑛=1

      34417  nn

    Answer

  • Problem 4: Finding the sum of a series What is the sum of the series written in summation notation.

    a)

    b)

     

      635,123538 2

    70 S

    aa 2

    n Sn

    70

    n1

    

    

    70 n terms, so70There are 

    1a Find   8315a1 

    70Find a   3533705a70 

    etic is arithmthe series function

    a linearformula isSince the

                 2222222 17161514131211 

    3625169410  91

      

    70

    1n

    3n5

      

    7

    1n

    2 1n

    Answer

  • Your turn

    Answers a) 2140 b) 100 c) 1

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