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Arithmetic Progression (A.P.)

The sequence < x1x2x3, ...., xn, .... > is called an Arithmetic progression (A.P.), if x2 – x1 x3 – x2 x4 – x3 = ...... = xn – xn–1 = ....... In general (xn – xn–1) = constant, n  N. This constant difference is called Common Difference . If 'a' is the first term and 'd' is the common difference, then the A.P. can be written as a + (a + d) + (a + 2d) + (a + 3d) + ...... + {a + (n – 1)d} + .....
Note that a, bc are in AP 2b = a + c
General Term of an AP:
General term (i.e. nth term) of an AP is given by
Tn = a + (n – 1) d


  1. If a sequence has n terms, then its nth term is also denoted by  which indicate last term
  2. Common difference can be zero, + ve or – ve.
  3. If there are n terms in an A.P, then mth term from end = (n – m + 1)th term from beginning.


Sum of first n terms of an AP:
The sum of first n terms of an A.P. is given by
Sn = Description: 615.png[2a + (n – 1) d] or Sn = Description: 620.png[a + Tn]
or Description: 625.png
Some standard results:
(i) Sum of first n natural numbers,
Description: 630.png=Description: 635.png
(ii) Sum of first n odd natural numbers,
Description: 640.png = n2
(iii) Sum of first n even natural numbers,
Description: 645.png = n (n + 1)
(iv) Sum of squares of first n natural numbers,
Description: 650.png=Description: 655.png 
(v) Sum of cubes of first n natural numbers,
Description: 660.png = Description: 665.png 

Arithmetic Mean (A.M)

The A.M. between the two given quantities a and b is A if a, A, b are in A.P.
i.e. A – a = b – A  A = Description: 670.png


A.M. of any n positive numbers a1a2, .........., an is
A = Description: 675.png

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