Introduction:
Solving online progression is very interesting since we can find the nth term of the particular sequence in much easier way. In this article we shall learn about steps involved in progressions solving. Moreover we will see in detail about different types involved in progression.
There are three types of Progression in math,
Let us see these progression and their properties in the following section.
Arithmetic Progression
Definition:
It is a sequence of numbers in which each term except the first term can be calculated by adding constant number (common difference) to the immediately preceding number.
The General form of the arithmetic sequence is,
a, a+d, a+2d, a+3d………..
Here a is the first number and d is the common difference.
To find the nth term of an arithmetic progression we can use the following formula,
an=a+ (n-1) d
Properties of Arithmetic Progression:
When we add or subtract any constant number with all the terms of the sequence, the arithmetic sequence remains an arithmetic sequence.
Example:
5, 7, 9, 11, 13, 15, 17….. is an A.P with common difference 2.
Add 3 with all the terms,
8, 10, 12, 14, 16, 18, 20…. Is also an A.P with common difference 2.
When we multiply or divide by a non-zero constant with all the terms of the sequence, the arithmetic progression sequence remains an arithmetic progression.
Example:
10, 20, 30, 40, 50, 60, 70…. Is an A.P with common difference 10.
Multiply by 2 with all the terms,
20, 40, 60, 80, 100, 120, 140…. Is also an A.P with common difference 20.
Example Problem:
1. Find the 6th term of the following sequence, 7, 12,17,22,27…
Solution:
The common difference is 5.
nth term of the A.P is an= a+(n-1)d
6th term of the A.P is a6 = 7+ (6-1)5
= 7+ 5 × 5
= 7 + 25
= 32
So the 6th term of the given sequence is 32.
Geometric Progression
Definition:
It is a sequence of numbers in which each term of the sequence except the first term can be calculated by multiplying the preceding term by means of a constant factor (common ratio).
The General form of the Geometric progress is,
a, ar,ar2,ar3,………
nth term of the geometric progression is,
an=ar (n-1)
Properties:
When we multiply or divide by a non-zero constant with all the terms of the sequence, the arithmetic progression sequence remains an arithmetic progression.
Ex: 2, 4, 8, 16, 32, 64…. is the geometric series with common ratio 2.
Multiply by 2,
4, 8,16, 32, 64, 128…. Is also the geometric series with common ratio 2.
Harmonic Progression:
Definition:
In mathematics, a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
The general form of the harmonic progression is ,
a , a , a , a .............
1+d 1+2d 1+3d
Example:
10, 10/6 , 10/11 , 10/16....
Here a = 10 and d = 5.
Solving online progression is very interesting since we can find the nth term of the particular sequence in much easier way. In this article we shall learn about steps involved in progressions solving. Moreover we will see in detail about different types involved in progression.
There are three types of Progression in math,
- Arithmetic progression
- Geometric progression
- Harmonic progression
Let us see these progression and their properties in the following section.
Arithmetic Progression
Definition:
It is a sequence of numbers in which each term except the first term can be calculated by adding constant number (common difference) to the immediately preceding number.
The General form of the arithmetic sequence is,
a, a+d, a+2d, a+3d………..
Here a is the first number and d is the common difference.
To find the nth term of an arithmetic progression we can use the following formula,
an=a+ (n-1) d
Properties of Arithmetic Progression:
When we add or subtract any constant number with all the terms of the sequence, the arithmetic sequence remains an arithmetic sequence.
Example:
5, 7, 9, 11, 13, 15, 17….. is an A.P with common difference 2.
Add 3 with all the terms,
8, 10, 12, 14, 16, 18, 20…. Is also an A.P with common difference 2.
When we multiply or divide by a non-zero constant with all the terms of the sequence, the arithmetic progression sequence remains an arithmetic progression.
Example:
10, 20, 30, 40, 50, 60, 70…. Is an A.P with common difference 10.
Multiply by 2 with all the terms,
20, 40, 60, 80, 100, 120, 140…. Is also an A.P with common difference 20.
Example Problem:
1. Find the 6th term of the following sequence, 7, 12,17,22,27…
Solution:
The common difference is 5.
nth term of the A.P is an= a+(n-1)d
6th term of the A.P is a6 = 7+ (6-1)5
= 7+ 5 × 5
= 7 + 25
= 32
So the 6th term of the given sequence is 32.
Geometric Progression
Definition:
It is a sequence of numbers in which each term of the sequence except the first term can be calculated by multiplying the preceding term by means of a constant factor (common ratio).
The General form of the Geometric progress is,
a, ar,ar2,ar3,………
nth term of the geometric progression is,
an=ar (n-1)
Properties:
When we multiply or divide by a non-zero constant with all the terms of the sequence, the arithmetic progression sequence remains an arithmetic progression.
Ex: 2, 4, 8, 16, 32, 64…. is the geometric series with common ratio 2.
Multiply by 2,
4, 8,16, 32, 64, 128…. Is also the geometric series with common ratio 2.
Harmonic Progression:
Definition:
In mathematics, a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
The general form of the harmonic progression is ,
a , a , a , a .............
1+d 1+2d 1+3d
Example:
10, 10/6 , 10/11 , 10/16....
Here a = 10 and d = 5.
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