Arithmetic Progressions

Let us study what is meant by Arithmetic Progressions,
Consider the following lists of numbers :
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii) – 3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
Each of the numbers in the list is called a term.
Given a term, can you write the next term in each of the lists above? If so, how
will you write it? Perhaps by following a pattern or rule. Let us observe and write the
rule.
In (i), each term is 1 more than the term preceding it.
In (ii), each term is 30 less than the term preceding it.
In (iii), each term is obtained by adding 1 to the term preceding it.
In (iv), all the terms in the list are 3 , i.e., each term is obtained by adding
(or subtracting) 0 to the term preceding it.
In (v), each term is obtained by adding – 0.5 to (i.e., subtracting 0.5 from) the
term preceding it.
In all the lists above, we see that successive terms are obtained by adding a fixed
number to the preceding terms. Such list of numbers is said to form an Arithmetic
Progression ( AP ).
So, an arithmetic progression is a list of numbers in which each term is
obtained by adding a fixed number to the preceding term except the first
term.
This fixed number is called the common difference of the AP. Remember that
it can be positive, negative or zero.
Now let us learn about finite arithmetic progression.