Arithmetic Progression
An arithmetic progression ( AP ) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant . For instance , sequence 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 … is an arithmetic progression with common difference of 2 . Each of the numbers in the list is called a term .
An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term . E . g .: 2,4,6,8,10 …. This fixed number is called the common difference of the AP . This common difference can be positive , negative or zero
o Positive common difference ( 3 ): 1,4,7,10,13 .. o Negative common difference ( -1) : 7,6,5,4,3,2,1,0 , -1 … o Zero Common Difference : 3,3,3,3,3,3,3,3,3,3 ….
Let us denote the first term of an AP by a1 , second term by a2 , . . ., nth term by an and the common difference by d . Then the AP becomes a1 , a2 , a3 , . . ., an .
So , a2 – a1 = a3 – a2 = . . . = an – an – 1 = d .
Thus a , a + d , a + 2d , a + 3d , . . . represents an arithmetic progression where a is the first term and d the common difference . This is called the general form of an AP .
AP with a finite number of terms are called a finite AP E . g .: { 1,3,5,7 }, while AP with infinite number of terms is called infinite AP , E . g .:{ 1,3,5,7 ,…}.
To know about an AP , the minimum information that we need is
1 . First term , denoted by ‘ a ’ 2 . Common difference , denoted by ‘ d ’
With this we can form the AP as a , a + d , a + 2d , a + 3d , ……
For instance if the first term a is 5 and the common difference d is 2 , then the AP is 5 , 7,9 , 11 , . . .