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 aaaaaa2nd 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 372

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 aa2

nS

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 aa2

nS

2550

10022

50S50

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

10309942

20S20

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 aa2

nS

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 aa2

nS

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

n2108642

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)

635,1235382

70S

aa2

nSn

70

n1

70 n terms, so70There are

1a Find 8315a1

70Find a 3533705a70

etic is arithmthe series function

a linearformula isSince the

222222217161514131211

3625169410 91

70

1n

3n5

7

1n

21n

Answer

Your turn

Answers a) 2140 b) 100 c) 1

Problem 5: Using a G.C.

From the LIST menu. (2nd STAT)

What is the sum of the series written in summation notation

n100

0n

4

1n

340

1n

1- c) n b)8n3a)

70

1n

3n5

50

1n

2 nn

Your turn

Answer: 41,650

Answer

Classwork odd Homework even