Progressions are numbers arranged in a particular order such that they form a predictable order. It means that from that series we can predict the next numbers of that series or sequence. Arithmetic Progression is the sequence of the numbers where the difference between any of the two consecutive numbers is the same throughout the sequence. It can also be the common difference of that series.
Four types of Sequence
There are mainly four types of sequences in Arithmetic, Arithmetic Sequence, Geometric Sequence, Harmonic Sequence, and Fibonacci Sequence. All four sequences are different and have unique relations among their terms. Let’s look at these 4 types of sequences in detail,
Arithmetic Sequence
An Arithmetic sequence is a sequence in which every term is created by adding or subtracting a definite number to the preceding number. The first term is represented as “a”, the common difference is denoted as “d”. Examples of Arithmetic sequences are,
Some examples of Arithmetic Sequence
- 1, 2, 3, 4, 5, 6,…
- 2, 2, 2, 2, 2, 2,…
- 22, 19, 16, 13, 10,…
Formulae for Arithmetic sequence
Arithmetic sequence
a, a+d, a+2d, a+3d, a+4d…
Or
a1, a2, a3, a4,… an
Nth term of the A.P.
an= a+ (n-1) d
Types of Arithmetic sequence
- Finite Sequence- Finite sequences have countable terms and do not go up to infinity. An example of a finite arithmetic sequence is 2, 4, 6, 8.
- Infinite Sequence- Infinite arithmetic sequence is the sequence in which terms go up to infinity. An example of an infinite arithmetic sequence is 2, 4, 6, 8,…
Geometric Sequence
A Geometric sequence is a sequence in which every term is created by multiplying or dividing a definite number to the preceding number. The first term of the geometric sequence is denoted as “a”, the common ratio is denoted as “r”.
Geometric sequence
a, ar, ar2, ar3, …. arn-1
Nth term of the G.P.
an= arn-1
Some examples of Geometric Sequence
- 2, 4, 8, 16, 32
- 2/3, 2/9, 2/27, 2/72,…
- 64, 32, 16, 8, 4, 2
Types of Geometric sequence
- Finite Sequence: Finite geometric sequence is the one in which terms are finite, an example of a finite geometric sequence is 64, 32, 16, 8, 4, 2.
- Infinite Sequence: Infinite geometric sequence is the one in which terms are infinite, example of an infinite geometric sequence is 2,4, 8, 16, 32, 64,…
Harmonic Sequence
A Harmonic sequence is a sequence in which the reciprocals of all the elements of the sequence form an arithmetic sequence and which can not be zero. A Harmonic progression’s first term is 1/a.
Example of Harmonic Sequence
1/2, 1/4, 1/6, 1/8, 1/10
Here reciprocal of all the terms is in the arithmetic sequence
2, 4, 6, 8, 10
Harmonic sequence
1/a, 1/b, 1/c,…
Nth term of the H.P.
an= 1/a+ (n-1)d
Fibonacci Sequence
Fibonacci Sequence is a special type of sequence of numbers in which each term is created by adding its previous two elements and the sequence starts with 0 and 1. The Fibonacci sequence can be defined as F0 = 0 , F1 = 1 and Fn = Fn-1 + Fn-2
Example of Fibonacci Sequence
0, 1, 1, 2, 3, 5, 8, 11, 19, …
Sample Problems
Question 1: Find the 40th term in the 2,5,8,11,14,… sequence
Solution:
a = 2
d = 5 – 2 = 3
n = 40
an = a + ( n -1 )d
a40 = 2 + ( 40 – 1 )3
= 2 + 117
a40 = 119
Question 2: Find the 9th term in the 1/3, 2/3, 1, 4/3… sequence
Solution:
a = 1/3
d = 2/3 – 1/3 = 1/3
n = 9
an = a + ( n -1 )d
a9 = 1/3 + ( 9 – 1 )(1/3)
= 1/3 + 8/3
a40 = 9/3 = 3
Question 3: Find the 7th term of 4,12,36,108,.. geometric sequence
Solution:
a = 4
r = 12/4 = 3
an = a.rn-1
a7 = 4 . 37 – 1
= 4 . 36
= 4 . 729
a7= 2916
Question 4: Find the 11th term of 2,4,8,16,.. geometric sequence
Solution:
a = 2
r = 4/2 = 2
an = a.rn-1
a11 = 2 . 211 – 1
= 2. 210
= 2 . 1024
a11= 2048
Question 5: Find the 6th term of 1/2, 1/4, 1/6, 1/8, .. harmonic sequence
Solution:
Here, the A.P. will be 2, 4, 6, 8, ..
So, a = 2
d = 4 – 2 = 8 – 6 = 2
n = 6
a6 = a + ( n – 1 ) . d
= 2 + ( 6 – 1 ) . 2
= 2 + 10
a6 = 12
Therefore the 6th term in the harmonic sequence would be 1/a6 = 1/12