Sequences
Unit 1:
Arithmetic Sequences and Series
• A sequence is a list of numbers that follow a pattern.
• The symbols t1, t2, t3, t4, ... are used to represent the terms of a sequence.
• The general term of a sequence can be represented by tn.
• The sequence t1, t2, t3, ..., tn
is called a finite sequence because it comes to a definite end.
• The sequence t1, t2, t3, ..., tn, tn+1, ... is called an infinite sequence because the sequences never ends.
Try This
Determine the next four terms of the following sequences.
Arithmetic ( Linear) Sequences
A pattern of numbers in which each term is found by adding or subtracting the preceding term by the same amount.
5, 8, 11, 14, ...
Each term is 3 more than the previous term.
17, 15, 13, 11, ...
Each term is 2 less than the previous term.
5, 8, 11, 14, ____, ____, ____, ____, ...
17, 15, 13, 11, ____, ____, ____, ____, ...
The first term is represented by a, and the difference between any two consecutive terms is called the common difference (d).
The general arithmetic sequence can be represented as follows:
a, a+d, a+2d, a+3d, ..., a+(n­1)d, ...
Graphs of Arithmetic Sequences
Arithmetic Formulas
What is the "slope" of each graph?
• A recursive (implicit) formula defines each term with reference to a previous term. The first term must be given in a recursive formula.
tn = tn­1 +d,
n>1, n∈N, t1=a
value
value
• A non­recursive (explicit) formula does not use previous terms.
tn = a +(n­1)d
term number
term number
1
Example
Example
Decide if each sequence is arithmetic.
Write a rule for the nth term of each sequence. Then find t25.
1)
­5,
­1,
3,
7,
2)
4,
5,
7,
10, 14, ...
3)
1,
4,
8,
12, 16, ...
4)
­4,
­8,
11, ...
­12, ­16, ­20, ...
1)
48, 53, 58, 63, ...
tn = t25 = t25 = 5(25) + 43
t25 = 125 + 43
t25 = 168
tn = a + (n - 1)d
tn = 48 + (n-1)5
tn = 48 + 5n - 5
tn = 5n + 43
Example
Example
Write a rule for the nth term of each sequence. Then find t25.
Write a rule for the nth term for the arithmetic sequence.
2)
tn = t25 = t15 = a + (14)d
10 = a + 14d
a = 10 - 14d
so.....
tn = -18n - 3
t20 = 25
t15 = 10,
­21, ­39, ­57, ­75, ...
t25 = -453
t20 = a + 19d
25 = a +19d
a = 25 - 19d
a=a
10-14d = 25-19d
5d=15
d=3
and
a = -32 (put d back into either equation)
tn = -32 +(n-1)3
So a rule could be: tn = 3n - 35
Another method (easier, maybe.....)
t15 = 10 t20= 25
Treat it like slope: y x
25 ­ 10 = 15 = 3
20 ­ 15 5
∴ d = 3 We still need to determine "a" :
t15 = a + 14(3)
a = -32
t15 = 10 = a + 42,
tn = 3n - 35
So tn = -32 + (n-1)3
2
Another type of question.......
Homework: For Sept 10
For the arithmetic sequence {19, 9, ­1, ­11, ...} which term has a value of ­221?
Worksheeet on Arithmetic Sequences
Solution:
1­8, 11, 12, 15, 16, 19, 20
tn = a + (n­1)d
­221 = 19 + (n­1)(­10)
­221 = 19 ­ 10n + 10
­221 = ­10n + 29 ­250 = ­10n
n = 25
Recursive formulas:
We can also work through problems in context......
A sequence is defined by the following rule:
Example 1: To repay the bank for a loan, a customer
agrees to increase each payment by $11.50. If the first
payment is $32.50, how much is the ninth payment?
t1 = 2, t2 = 5, tn = tn­2 + tn­1 , n> 1, n∈N,
Find t4.
a = $32.50
d = $11.50
tn = a + (n - 1)d
Solution:
t9 = 32.50 + (9-1)11.50
t4 = t4­2 + t4­1
t9 = 32.50 + 8(11.50)
t4 = t2 + t3
t9 = 32.50 + 92
t9 = $124.50 so the ninth
payment would be $124. 50
Need to find t3......t3= t1 + t2
=2+5
=7
t4 = t2 + t3 = 5 + 7 = 12
Series
A series is the sum of the terms of a sequence. If tn is a sequence, then the sum of the first n terms of the sequence is Sn.
Worksheet ­ Arithmetic Sequences
S1 = t1
S2 = t1 + t2
S3 = t1 + t2 + t3
Sn = t1 + t2 + t3 + ... + tn
sequence
a, b, c, d, ...
series
a + b + c + d + ....
3
Example
Sum of a Finite Arithmetic Series
Find the sum of the integers from 1 to 100.
1 + 2 + 3 + 4 + .... + 97 + 98 + 99 + 100 = ?
Trick:
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
.
.
.
so, 50 pairs X 101 = 5050
Example
Example
Find the sum of the first 20 terms for;
Find the sum of the first 50 terms for;
12 + 18 + 24 + 30 + 36 + ...
Sn = n/2(2a + (n - 1)d)
34 + 45 + 56 + 67 + 78 + ...
Sn = n/2(2a + (n - 1)d)
S20 = 20/2[2(12) + 19(6)]
S50 = (50/2)[2(34) + 49(11)]
S20 = 10[24 + 114]
S50 = 25[68 + 539]
S20 = 10(138)
S50 = 25[607]
S20 = 1380
S50 = 15175
Example
Find the sum of
Example
5 + 9 + 13 + ... + 85
Sn = n/2[t1 + tn]
...need to find what "n" is .... look at our general formula
Find the sum of
5 + 9 + 13 + ... + 85
Sn = n/2[t1 + tn]
...need to find what "n" is .... look at our general formula
tn = a + (n - 1)d
85 = 5 + (n - 1)4
85 = 5 + 4n - 4
tn = a + (n - 1)d
85 = 5 + (n - 1)4
85 = 5 + 4n - 4
85 = 4n + 1
84 = 4n
85 = 4n + 1
84 = 4n
n = 21
n = 21
S21 = 21/2[ 5 + 85]
S21 = 21/2(90)
S21 = 945
4
Homework for Sept 13:
Arith. Seq: # 27, 29
Arith Series: #1,2,11,15
Find "n" if Sn = 20 for the following series:
­16 + (­12) + (­8) + (­4) + 0 + ...
Sigma Notation ( Σ )
Sn = n (2a + (n ­ 1) d) Now substitute in what you know...
2
20 = n(2(­16) + (n ­ 1)4) Now simplify...
2
The Greek letter Σ (sigma) corresponds to the English letter S and stands for sum. Sigma notation is often used to abbreviate the writing of a series.
For example, to abbreviate the series:
2 + 4 + 6 + 8 + 10 tn = 2n for the five terms
20 = n(­32 + 4n ­ 4)
2
20 = n(4n ­ 36)
2
Now divide by two
20 = n(2n ­ 18)
20 = 2n2 ­ 18n
0 = 2n2 ­ 18n ­ 20
Now we have a quadratic so find roots...
0 = n2 ­ 9n ­ 10
The variable k is called the index of summation, and it tells you what terms to use. The lower limits indicates the initial term to start with, and the upper limit indicates what term to end with. In this example we used all of the terms from 1 to 5.
0 = (n­10)(n + 1) so, n = 10 or n =­1 .........so n = 10
Example
Write each series in expanded form and then find the sum.
b)
a)
a)
= (3(1) +2) + (3(2) + 2) + (3(3) + 2) + (3(4) + 2) + (3(5) + 2) + (3(6) + 2)
=5 + 8 + 11 + 14 + 17 +20
= 75
b) = (-3 + 5) + (-4 + 5) + (-5 + 5) + (-6 + 5) + (-7 + 5) + (-8 + 5) + (-9 +5)
= 2 + 1 + 0 + -1 + -2 + -3 + -4
= -7
5
6