Arithmetic progression (A. P.):

An A.P. is a sequence whose terms increase or decrease by a fixed number, defined as the common difference of the A.P.

n^th Term and  Addition of n Terms:

If a is the first term and d the general difference, the A.P. can be known as a, a + d, a + 2d, .... The nth term  a_n is given by a_n = a + (n - 1)d.

The sum S_n of the first  n^th terms of such an A.P. is shown by (a + l ) where l  is the last term (i.e. the nth term of the A.P.).

Notes

• If a fixed number is subtracted or added to each term of a provided A.P. then the resulting linear sequence is also an A.P. with the similar common difference as that of the provided A.P.
• If each and every term of an A.P. is product by a fixed number(say k) (or divided by a non-zero fixed number), the resulting linear sequence is also an A.P. with the common difference multiply by k.
• If a₁, a₂, a₃.....and b₁, b₂, b₃...are two A.P.'s with basic differences d' and d respectively then a₁+b₁, a₂+b₂, a₃+b₃,...is also an A.P. with basic difference d+d'
• If we have to take three terms in an A.P., it is convenient to get them as a - d, a,
  a + d. Usually, we take a - rd, a - (r - 1)d,......a - d, a, a + d,.......a + rd in that case we have to take (2r + 1) related terms in an A.P
• If we have to take four parts, we take a - 3d, a - d, a + d, a + 3d. In general, we take
  a - (2r - 1)d, a - (2r - 3)d,....a - d, a + d,.....a + (2r - 1)d, in that type we have to take 2r terms in an A.P.
• If a₁, a₂, a₃, ....... a_n are in A.P. then a₁ + a_n = a₂ + a_n-1 = a₃ + a_{n -2} = . . . . . and so on.

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