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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 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