1. Censure
By Saturnino C. Guardiario Jr.
I am a man in the wilderness looking for the truth. I was born with disability. I was
born incomplete. But who shall I blame? I was born in this world because of some reasons. I
was born incompletely because nobody knows ( excluding the Supreme Being ) my destiny.
Unlike the animals , we are already aware of their destination. They eat grasses, meat or
both in their entire lives. Who shall we blame?
“Sequence and Series”
Many situations in my real life involves sequence and series such as looking at the
time, weekly visitation of the collector of the money to those who loaned money from their
company and many more.
While I am studying, I learned different kind of sequences and series which are tools
to awaken myself and to wide open my eyes in the happening of our community today.
Some sequences and series include arithmetic sequences and series, geometric sequence
and series, harmonic sequence and Fibonacci sequence.
The first series and sequence I learned was arithmetic sequence .
“Arithmetic Sequence”
Arithmetic sequence is a sequence in which each term after the first is obtained by
adding a constant d to the preceding term where the constant number d is called the
common difference.
For instance :
If a1 the first term and d is the common difference, then the terms of an
arithmetic sequence can be enumerated in the following manner:
A1 = the first term
A2 = a1 + d
A3 = a1 + 2d
A4 = a1 + 3d
A5 = a1 + 4d
A6 = a1 + 5d
“Finding the common difference of the Arithmetic Sequence”
Consider the following sequence and observe how the succession of the terms is obtained.
20, 30, 40, 50,
30 – ( 20 ) = 10
40 – ( 30 ) = 10
50 – ( 40 ) = 10
This show that the coefficient of d is one less than the number
of terms. In general, an = a1 + ( n – 1 ) d
Where: a1 = the first term d = the common difference
An = the nth term
The common difference is 10.
Remember :
The nth term of an arithmetic sequence is defined as an = a1 +
(n – 1) d
Where: a1 = the first term
a1 = the nth term
d = the common difference

2. The second sequence and series I learned was Arithmetic Series. Arithmetic series
is a series whose associated sequence is arithmetic. Series is the indicated sum of the terms
of a sequence. The word “series “ can be used both in the singular and in the plural form.
For the arithmetic expression of the form a1 + ( a1 + d ) + (a1 + 2d ) + . . . + [ a1 + ( n –
1 )d ] is called an arithmetic series.
For example :
You were asked to find the sum of the first 10 terms of the arithmetic sequence 4, 10, 16 .
. .
Given : a1 = 4 d = 6
n = 10 Sn = ?
Using the formula : Sn =
𝑛
2
[ 2a1 + ( n – 1 ) d ]
Sn =
10
2
[ 2 ( 4 ) + ( 10 – 1 ) 6 ]
Sn = 5 ( 8+ 54 )
Sn = 5 ( 62 )
Sn = 310
“Geometric Sequence”
The third Sequence and Series I learned was Geometric Sequence. Like the
arithmetic sequence , each of the terms in a geometric sequence is related to the preceding
term through a definite pattern.
A geometric sequence is a sequence in which a term is obtained by multiplying the
preceding term by a constant number r , called the common ratio.
Finding the common ratio of the geometric sequence
Take a look how the common ratio is obtained. Using the example 2, 4, 8, 16, 32 . . .
𝑠𝑒𝑐𝑜𝑛𝑑 𝑡𝑒𝑟𝑚
𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
=
4
2
= 2
𝑓𝑜𝑢𝑟𝑡ℎ 𝑡𝑒𝑟𝑚
𝑡ℎ𝑖𝑟𝑑 𝑡𝑒𝑟𝑚
=
16
8
= 2
𝑓𝑖𝑓𝑡ℎ 𝑡𝑒𝑟𝑚
𝑓𝑜𝑢𝑟𝑡ℎ 𝑡𝑒𝑟𝑚
=
32
16
= 2
Remember:
The formula for the sum of the 1st n terms in an
arithmetic sequence is
Sn =
𝑛
2
( a1 + an ) or Sn =
𝑛
2
[ 2a1 + ( n – 1 ) d ]
Where : Sn = the sum d = common difference
a1 = first term an = nth term
Therefore, the common ratio is obtained
through dividing any terms by the preceding
term of it.

3. Finding the nth
term of a Geometric Sequence
Given the first and the common ratio of a geometric sequence, nth term can be found
using the formula an = a1r n – 1 .
For instance:
You were asked to find the tenth term of the geometric sequence 2, 4, 8, . . .
Given: a1 = 2
r = 2 n = 10
Using the formula an = a1rn-1
A10 = 2( 2 )10-1
A10 = 2( 2 )9
A10 = 2( 512 )
A10 = 1024  the tenth term
“Geometric Series”
The fourth sequence and series I learned was Geometric Series. A geometric series
can be found much the same way as the arithmetic series. A corresponding formula can be
worked out. Geometric series is a series whose associated sequence is geometric.
The Finite Geometric Series
The finite geometric series is geometric series which has a beginning and an ending.
For example :
You were asked to find the sum of the geometric series 40 + ( - 20 ) + 10 + ( - 5 ) +
5
2
+ (-
5
4
)
Given: a1 = 40 n = 6 r = -
1
2
Using the formula:
Remember :
The nth term of a geometric sequence is an =
a1rn-1
Where : a1 = the first term
an = the nth term
r = the common ratio
Remember :
The formula for the sum of the first n terms in
a geometric sequence is Sn =
𝑎1−𝑎1𝑟𝑛
1 – r
or
Sn =
𝑎1 (1−𝑟𝑛)
1 – r
Where: Sn = the sum r = the common ratio,
r ≠ 1
a1 = the first term
Sn =
a1 – a1rn
1− 𝑟
Sn =
40 – 40(−
1
2
)6
1−(−
1
2
)
Sn =
40 – 40(
1
64
)
1
1
2
Sn =
40 –
5
8
3
2
Sn =
39
3
8
3
2
Sn = 39
3
8
÷ -
3
2
 39
3
8
.
2
3

315
8
.
2
3
 26
1
4

4. Infinite Geometric Series
The fifth sequence and series I learned was infinite geometric series. Infinite geometric
series is the indicated sum of the terms of an infinite geometric sequence. The series 1 + 2 +
4 + 8 + 16 + . . . is an example of an infinite geometric series. If the terms of a geometric
sequence is a1 and the common ratio is r, then the infinite geometric series S is Sn = a1 + a1r
+ a1r2 + . . . Literally, an infinite series is only a symbolic expression. A series is given precise
meaning through the notion of limits, a basic concept in Calculus. Its discussion is beyond
the scope of this masterpiece.
For example:
Find the sum of
3
10
+
3
100
+
3
1000
+ . . .
a1 =
3
10
r =
1
10
Using the formula Sn =
𝑎1
1−𝑟
We have
Sn =
3
10
1−
1
10
Sn =
3
10
9
10
Sn =
3
10
÷
9
10

3
10
.
10
9

30
90
Sn =
1
3
Remember :
The sum of the terms of an infinite
geometric sequence is Sn =
𝑎1
1−𝑟
As I learned that these are the formulas in finding the
sum of the terms of a geometric sequence.
For finite sequence:
a. Sn =
𝑎1−𝑎1𝑟𝑛
1−𝑟
 for r ≠ 1
b. Sn = a1n  for r = 1
c. Sn = 0  for r = -1 and n is even
d. Sn = a1  for r = -1 and n is odd
For the sum to infinity:
Sn =
𝑎1
1−𝑟
for 1 < r < 1

5. “Harmonic Sequence”
The sixth sequence and series I learned was harmonic sequence. According to it that a
sequence of number is said to be harmonic if their reciprocals form an arithmetic sequence.
The sequence
1
3
,
1
6
,
1
9
,
1
12
is a harmonic sequence since their reciprocals 3, 6, 9, 12 form an
arithmetic sequence.
In general, a harmonic sequence may be represented as
1
𝑎1
,
1
𝑎1+𝑑
,
1
𝑎1+2𝑑
,
1
𝑎1+3𝑑
…
1
𝑎1+( 𝑛−1) 𝑑
Examples of harmonic sequence together with the corresponding arithmetic sequence are
given as follows:
Illustrative examples :
Find the first six terms of a harmonic sequence where a1 is
1
3
and d is
1
5
.
The corresponding arithmetic sequence with a1 = 3 and d = 5 will be 3, ,8 ,13, 18, 23, 28 . . .
Therefore, the harmonic sequence is
1
3
,
1
8
,
1
13
,
1
18
,
1
23
,
1
28
. . .
“Fibonacci Sequence”
The last but not the least sequence I learned was Fibonacci sequence. A special
relationship among the terms of a particular sequence was discovered by Leonardo
Fibonacci of Pisa. This is known as Fibonacci sequence. The sequence recursively defined
by Fn+2 = Fn + Fn+1 for n ≥ 1 where F1 = 1 F2 = 1
For example:
1, 1, 2, 3, 5, 8, 13, 21, 34 . . .
The relation was noted in an experiment using a pair of male and female rabbits.
The pair of rabbits produced a new pair of rabbits after two months, and every month
Arithmetic Sequence
3
2
,
7
2
,
11
2
,
15
2
,…
2, 4, 6, 8, 10, . . .
Harmonic Sequence
2
3
,
2
7
,
2
11
,
2
15
,…
1
2
,
1
4
,
1
6
,
1
8
,
1
10
,…

6. thereafter. In like manner, the new pair of rabbits produced a pair after 2 months and so
on.
Analyzing the relationship where R represents the pair of rabbits, m the month of
the birth of the pair, gives the total number of pairs of rabbits.
Month
1st
2nd
3rd
4th
5th
6th
Growth in pairs of rabbits
R
R
R + R3
R +R3 +R4
R +R3+R4+R5+R3-5
R+R3+R4+R5+R6+R3-5+R3-6+R4-6
Total no. of pairs
1
1
2
3
5
8
R3-5 meansthe rabbits born on the 3rd monthR3 now producesapair of rabbits born on
the 5th month. The sequence formed is1, 1, 2, 3, 5, 8, 13 respectively.
The spiralchambers in the shell
of the nautilusclearly exhibit the
Fibonacci sequence.
The beauty of the sequence
or pattern in our daily lives must
be appreciated by us. Take good
care of it and continue explore
and go deeper. Do not stay on the
same place. Go out of your
comfort zone and discover the
excellence and the magnificence
of the handmaid of the Lord.
After doing it, now who
shall we blame?