. Arithmetic Progression Let us begin with the following two illustrations. Illustration Make the following figures using match sticks (i) How many match sticks are required for each figure? , , and .
(ii) Can we find the difference between the successive numbers? Therefore, the difference between successive numbers is always . Fig. .
Illustration A man got a job whose initial monthly salary is fixed at ₹ , with an annual increment of ₹ . His salary during st , nd and rd years will be ₹ 10000 , ₹ 12000 and ₹ 14000 respectively. If we now calculate the difference of the salaries for the successive years, we get 12000 10000 14000 12000 – ; – . Thus the difference between the successive numbers (salaries) is always .
Did you observe the common property behind these two illustrations? In these two examples, the difference between successive terms always remains constant. Moreover, each term is obtained by adding a fixed number ( and in illustrations and presented above) to the preceding term except the first term . This fixed number which is a constant for the differences between successive terms is called the “common difference” .
Definition Let a and d be real numbers. Then the numbers of the form a , a , a + , + , a + , ... is said to form Arithmetic Progression denoted by A.P. The number ‘ a ’ is called the first term and ‘ d ’ is called the common difference .
Simply, an Arithmetic Progression is a sequence whose successive terms differ by a constant number. Thus, for example, the set of even positive integers , , , , , ,… is an A.P. whose first term is a = and common difference is also d = since = , = , = , … Most of common real−life situations often produce numbers in A.P.