Contributed by:
DISCUSSION POINTS:
1. What is a sequence?
2. What is the difference between finite and infinite?
1.
UNIT IV & UNIT V
SEQUENCES AND
SERIES
Dr. T. VENKATESAN
Assistant Professor
Department of Statistics
St. Joseph’s College, Trichy-2.
2.
• What is a sequence?
• What is the difference between
finite and infinite?
3.
Sequence:
• A function whose domain is a set of consecutive
integers (list of ordered numbers separated by
commas).
• Each number in the list is called a term.
• For Example:
Sequence 1 Sequence 2
2,4,6,8,10 2,4,6,8,10,…
Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5
Domain – relative position of each term (1,2,3,4,5)
Usually begins with position 1 unless otherwise
stated.
Range – the actual “terms” of the sequence
(2,4,6,8,10)
4.
Sequence 1 Sequence 2
2,4,6,8,10 2,4,6,8,10,…
A sequence can be finite or infinite.
The sequence has The sequence
a last term or final continues without
term. stopping.
(such as seq. 1) (such as seq. 2)
Both sequences have an equation or general rule: an
= 2n where n is the term # and an is the nth term.
The general rule can also be written in function
notation: f(n) = 2n
6.
Write the first six terms of f (n) = (– 3)n – 1.
f (1) = (– 3)1 – 1 = 1 1st term
2nd term
f (2) = (– 3)
2–1
= –3
f (3) = (– 3)3 – 1 = 9 3rd term
f (4) = (– 3)4 – 1 = – 27 4th term
f (5) = (– 3)5 – 1 = 81 5th term
f (6) = (– 3)6 – 1 = – 243 6th term
You are just substituting numbers into
the equation to get your term.
7.
Examples: Write a rule for the nth term.
2 2 2 2
a. , , , ,... b. 3,5,7,9,...
5 25 125 625
2 2 2 2
1
, 2 , 3 , 4 ,...
5 5 5 5
Look for a pattern…
8.
Example: write a rule for the nth term.
9.
Describe the pattern, write the next term, and
write a rule for the nth term of the sequence
(a) – 1, – 8, – 27, – 64, . . .
SOLUTION
a. You can write the terms as (– 1)3, (– 2)3, (– 3)3,
(– 4)3, . . . . The next term is a5 = (– 5)3 = – 125.
A rule for the nth term is an 5 (– n)3.
10.
Describe the pattern, write the next term, and
write a rule for the nth term of the sequence
(b) 0, 2, 6, 12, . . . .
b. You can write the terms as 0(1), 1(2), 2(3),
3(4), . . . .
The next term is f (5) = 4(5) = 20. A rule for the
nth term is f (n) = (n – 1)n.
11.
Graphing a Sequence
• Think of a sequence as ordered pairs for
graphing. (n , an)
Term # Actual term
• For example: 3,6,9,12,15
would be the ordered pairs (1,3), (2,6),
(3,9), (4,12), (5,15) graphed like points in a
scatter plot. DO NOT CONNECT ! ! !
* Sometimes it helps to find the rule first
when you are not given every term in a
finite sequence.
13.
Retail Displays
You work in a grocery store and are stacking
apples in the shape of a square pyramid with 7
layers. Write a rule for the number of apples in
each layer. Then graph the sequence.
SOLUTION First Layer
STEP 1 Make a table showing the number of fruit
in the first three layers. Let an represent
the number of apples in layer n.
14.
STEP 2 Write a rule for the number
of apples in each layer.
From the table, you can see
that an = n2.
STEP 3 Plot the points (1, 1), (2, 4),
(3, 9), . . . , (7, 49). The graph
is shown at the right.
15.
• What is a sequence?
A collections of objects that is ordered so that
there is a 1st, 2nd, 3rd,… member.
• What is the difference between finite and
infinite?
Finite means there is a last term. Infinite
means the sequence continues without
stopping.
16.
Sequences and Series Day 2
• What is a series?
• How do you know the difference between a
sequence and a series?
• What is sigma notation?
• How do you write a series with summation
notation?
• Name 3 formulas for special series.
17.
Series
• The sum of the terms in a sequence.
• Can be finite or infinite
• For Example:
Finite Seq. Infinite Seq.
2,4,6,8,10 2,4,6,8,10,…
Finite Series Infinite Series
2+4+6+8+10 2+4+6+8+10+…
18.
Summation Notation
• Also called sigma notation
(sigma is a Greek letter Σ meaning “sum”)
The series 2+4+6+8+10 can be written as:
5 i goes from 1
2i
1
to 5.
i is called the index of summation
(it’s just like the n used earlier).
Sometimes you will see an n or k here instead of i.
The notation is read:
“the sum from i=1 to 5 of 2i”
19.
Summation Notation
Upper limit of summation
2i
Lower limit of summation
20.
Summation Notation for an
Infinite Series
• Summation notation for the infinite series:
2+4+6+8+10+… would be written as:
2i
1
Because the series is infinite, you must use i
from 1 to infinity (∞) instead of stopping at
the 5th term like before.
21.
Examples: Write each series using
summation notation.
a. 4+8+12+…+100 1 2 3 4
b. ...
2 3 4 5
• Notice the series can
be written as: • Notice the series
can be written as:
1 2 3 4
Or 4(i) where i goes ...
1 1 2 1 3 1 4 1
from 1 to 25.
i
25
Or, where i goes from 1 to .
i 1
4i
1
i
1 i 1
22.
Write the series using summation notation.
a. 25 + 50 + 75 + . . . + 250
SOLUTION
a. Notice that the first term is 25(1), the second
is 25(2), the third is 25(3), and the last is
25(10). So, the terms of the series can be
written as: ai = 25i where i = 1, 2, 3, . . . , 10
The lower limit of summation is 1 and the
upper limit of summation is 10.
ANSWER 10
The summation notation for the series is 25i.
i=1
23.
Write the series using summation notation.
b. 1 + 2 + 3 + 4 . . .
2 3 4 5
b. Notice that for each term the denominator
of the fraction is 1 more than the
numerator. So, the terms of the series can
be written as:
i
ai = i + where i = 1, 2, 3, 4, . . .
1
The lower limit of summation is 1 and the
upper limit of summation is infinity.
8
The summation notation for the series is i
i + 1.
i=1
24.
Example: Find the sum of the
10
2
series.
k
5
1
• k goes from 5 to 10.
• (52+1)+(62+1)+(72+1)+(82+1)+(92+1)+(102+1)
= 26+37+50+65+82+101
= 361
25.
Find the sum of the series.
(3 + k2) = (3 + 42) 1 (3 + 52) + (3 + 62) + (3 + 72) + (3 + 82)
= 19 + 28 + 39 + 52 + 67
= 205
26.
Find the sum of series.
7
11. (k2 – 1)
k=3
We notice that the Lower limit is 3 and the upper
limit is 7.
7
(k2 – 1)
k=3 = 9 – 1 + 16 – 1 + 25 – 1 + 36 – 1 + 49 – 1
= 8 + 15 + 24 + 35 + 48.
= 130 .
ANSWER 130.
27.
Special Formulas (shortcuts!)
nn n
n(n 1)
c1cn
n i
i
i1
1 i 1 2
n
2 n(n 1)(2n 1)
i
i 1 6
Page 437
28.
Example: Find the sum.
10
2
i
i 1
• Use the 3rd shortcut!
n(n 1)(2n 1) 10(10 1)(2 *10 1)
6 6
10 *11 * 21 2310
385
6 6
29.
Find the sum of series.
34
12. 1
i=1
We notice that the Lower limit is 1 and the
upper limit is 34.
34
1= 34.
i=1
ANSWER ... Sum of n terms of 1
34
1 = 34.
i=1
30.
Find the sum of series. Sum of first n positive integers is.
6
13. n n n (n + 1)
n=1 i =
i=1 2
6 (6 + 1)
We notice that the Lower =
2
limit is 1 and the upper limit
6 (7)
is 6. =
6 2
n = 1+2+3+4+5+6
42
= 21. =
2
or
= 21
ANSWER = 21
31.
• What is a series?
A series occurs when the terms of a sequence are
added.
• How do you know the difference between a
sequence and a series?
The plus signs
• What is sigma notation?
• How do you write a series with summation
notation?
Use the sigma notation with the pattern rule.
• Name 3 formulas for special series.
n
n n
n(n 1) n(n 1)(2n 1)
c cn
i 1
i 1
i
2
i2
i 1 6
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