[LEC]1.2 Geometric Series
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Transcript of [LEC]1.2 Geometric Series
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Business MathematicsDe La Salle University
Sequence
Geometric Sequence
Geometric Series
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Business MathematicsDe La Salle University
A sequence is the range of any function
from the set or a subset of natural
number to the set of real numbers.
SEQUENCE
To put it simply, it is an enumeration of real
numbers whose terms are define by a
specified pattern or formula:
naaaaa ,...,,,, 4321
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Each range of a given sequence is called
a term. If there are only finite number of
terms, then the sequence is called a
finite sequence, otherwise, it is called an
infinite sequence.
FINITE and INFINITE SEQUENCE
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Example:
1, 2, 3, 4, 5,
-each term is determined by adding 1 to
the previous term.
1/2, 2/3, 3/4, 4/5,
- the next term is determined by addingone to each numerator and denominator
of the previous term.
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Example:
0, 1, 2, 3, 5, 8, 13,
Fibonacci - each term is determined by
adding the two preceding terms
2, 3, 5, 7, 11, 13, 17, 19, 23,
- sequence of prime numbers
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A sequence whose terms, except the
first, are determined by multiplying a
fixed number to the term that precedes
it. The fixed number is called the
common ratio of the geometric
sequence.
GEOMETRIC SEQUENCE
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a1 = 1st term
a2 = 2nd term
an = nth term
n = number of terms
r = common ratio
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Example:
1, 2, 4, 8, 16, 32, 64, 128, 256
a1 = 1
a2 = 2
n = 9
r = 2a9 = 256
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Example:
27, -9, 3, -1, 1/3, -1/9, 1/27
a1 = 27
n = 7
r = -1/3
a7 = 1/27
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Given a geometric sequence:
a1, a2, a3, , anIf r is the common ration, then the
sequence can be written as:
a1, a1r, a1r2, , a1r
n-1
Henceand
GEOMETRIC SEQUENCE
1
1
!n
nraa
1
!
n
n
a
a
r
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1. Determine the required quantity giventhe geometric sequence: 2, 6, 18,
a. The common ration r.
b. The 5th term.
c. The 10th term.
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2. The 3rd and the fourth term of ageometric sequence are 20 and -40
respectively:
a. What is the common ration r.
b. Determine the 1st term.
c. Give the 10th term.
d. Which term is 20 480?
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3. The 3rd and the 6th term of a geometricsequence are 8/27 and -1 respectively:
a. What is the common ration r.
b. Determine the 1st term.
c. Which term is -729/64?
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Given a geometric sequence:
a1, a2, a3, , anThe indicated sum of the fist nth term is
called its geometric series.
a1 + a2 + a3 ++ an
GEOMETRIC SERIES
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Given a geometric series:
a1 + a2 + a3 ++ anwe have, S1 = a1, S2 = a1 + a2
Sn = a1 + a2 ++ an
GEOMETRIC SERIES
rraa
n
!
1
11
r
raa n
!
1
1
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Given a geometric series:
a1 + a2 + a3 +
If |r < 1|, then the value of thegeometric series is
INFINITE GEOMETRIC SERIES
r
aS
!
1
1
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1. Determine the required quantity giventhe geometric sequence: 1, 1/2, 1/4,
a. The common ration r.
b. The sum of the first 10 terms?
c. The value of S.
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2. The 4th and the 7th term of a geometricsequence are -0.4 and 0.0032
respectively:
a. What is the common ration r.
b. Determine the 1st term.
c. Find S20
.
d. What is the value of S?
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3. How do you write 1.232323 as afraction?