Arithmetic Progression (A.P.)



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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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