[lec]1.2 geometric series

8/8/2019 [LEC]1.2 Geometric Series

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Business MathematicsDe La Salle University

Sequence

Geometric Sequence

Geometric Series


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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:



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:



c. Find S20

.

d. What is the value of S?


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3. How do you write 1.232323 as afraction?

[lec]1.2 geometric series

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