Referred to as the sequence of number in a considerable order, you may observe that we very often come across arithmetic problems in regular lives. These considerable patterns of sequences and series have been addressed as progressions in mathematics.
Definition of Arithmetic Progression
A progression could generally be expressed as a special sequence for which you could possibly obtain a formula for solving out the nth term. Arithmetic Progressions are known to be the most commonly used sequences in mathematics with formulae that are easy to understand. Consider the below mentioned definitions:
Definition-1: It could be expressed as a mathematical sequence where the difference between two consecutive numbers is always supposed to be a constant and is known to be abbreviated as AP.
Definition-2: An arithmetic progression or sequence could be defined as a sequence of numbers where for each pair of consecutive term, the second number is achieved by adding up a fixed number to the previous one.
Definition-3: the fixed number that needs to be added to any element of AP in order to obtain the next element has been defined as the common difference of AP.
Common Difference of AP
For instance, consider the sequence, 1, 4, 7, 10, 13, 16,… it could be quite clearly regarded as a sequence with the common difference 3.
Generally, the common difference “d” for any arithmetic progression could be obtained as below:
x2 – x1 = x3 – x2 = …… = xn – xn-1 = d . Here, “d” is the common difference.
Any arithmetic progression could also be written as per the terms of common difference, as below:
x, x + d, x + 2d, x + 3d, x + 4d,…. ,x + (n – 1)d. Here, x is supposed to be the first term of the progression.
In order to find the nth term of a progression and also the sum of n terms included in an arithmetic series, the arithmetic formula has been given below:
For an arithmetic progression with n terms:
an = a + (n – 1) x d
a denotes the first term of the progression
d is the common difference
n is the number of terms, and;
an represents the nth term
It is also important to note that
- the finite portion of AP is referred as finite AP. Also, this is one of the reasons why the sum of a finite AP is addressed as arithmetic series.
- The behavior of any progression is dependent on the common difference.
- The member elements tend to grow towards positive infinity when the element “d” is positive.
- Similarly, the member terms tend to grow towards negative infinity if “d” is negative.
Above-mentioned are several basic details about Arithmetic Progression.