# Arithmetic Progression (A.P.)

#### notes

A sequence a_1, a_2, a_3,…, an,… is called arithmetic sequence or arithmetic progression if a_(n + 1) = a_n + d, n ∈ N, where a_1 is called the first term and the constant term d is called the common difference of the A.P.
The n^(th) term (general term) of the A.P. is a^n = a + (n – 1) d.

The sum to n term of A.P is S_n= n/2[2a+(n-1)d]

We can also write, S_n = n/2[a+l]

We can verify the following simple properties of an A.P. :
(1) If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
(2) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
(4) If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
(5) If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.

Arithmetic mean:

Given two numbers a and b. We can insert a number A between them so that a, A, b is an A.P. Such a number A is called the arithmetic mean (A.M.) of the numbers a and b. Note that, in this case, we have
A – a = b – A,    i.e., A  =(a+b)/2
We may also interpret  the A.M. between two numbers a and b as their average (a+b)/2.
For example, the A.M. of two numbers 4 and 16 is 10. We have, thus constructed an A.P. 4, 10, 16 by inserting a number 10 between 4 and 16.
The Arithmetic mean is d = (b - a)/(n + 1)

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