Arithmetic progression is that succession which has a persistent difference continued between the successive terms of the sequence. The representation of an Arithmetic Progression is the simplest of all the progressions. For instance, if the first term of the series is given as m and the common difference as 'r', then complete arithmetic progression can be written as follows:
m, m + r, m + 2 r, m + 3 r, m + 4 r, and so on to m + n r........... equation 1.
Where, 'n' is the total number of terms in arithmetic sequence. Let us consider an example of arithmetic succession to understand it better:
Example: If 11,x,5 are the terms of an AP, what is the value of x?
Solution: As we know the meaning of an arithmetic progression and the representation, that is given as shown in the equation 1. We can write the following results:
(m + r) – m = (m + 2 r) – (m + r) Consecutive terms have same difference between them.
Here in example, the given arithmetic sequence 11, x, 5 has first term as 11, x as the 2 nd term and 5 as 3rd term. Using the property of common difference between successive terms we can write:
x – 11 = 5 – x,
Taking the known quantities to right side of the equation and unknowns to other we get:
2 x = 5 + 11,
Or 2 x = 16,
Or x = 16 / 2 = 8,
So our arithmetic progression is: 11, 8, 5 with a constant difference of - 3 between the consecutive terms. Arithmetic progression finally obtained can be verified by checking the difference between the consecutive terms.