Arithmetic Progression: The sequence whose common difference between the two successive terms are same is called arithmetic sequence.
For example: 3,7,11,15

Let a be the 1st term of the sequence. Then, we know that the consecutive terms of arithmetic sequence have a common difference(d). Therefore, each term increases/decreases by “d”.

So,

  2nd term=(a+d)                       =a+(2-1)d

  3rd term=(a+d)+d   =a+2d =a+(3-1)d

  4th term=(a+2d)+d =a+3d =a+(4-1)d

  ……………………………………………………………
  ……………………………………………………………

  ……………………………………………………………

                     Therefore, nth term=a+(n-1)d

Hence, general term of Arithmetic Progression is given by;

     →  tn=a+(n-1)d

           where,

                  tn= nth term of sequence
                  a  = first term of sequence
                  n  = nth term of sequence
                  d  = common difference of sequence

Arithmetic mean: The terms in between the first and last term in the arithmetic progression are called arithmetic means.
Example; i) 2, 4, 6                   ii) 4, 8, 12, 16, 20, 24

    Here, the highlighted terms are the arithmetic means.

Formulas related to Arithmetic Means:

•The single arithmetic mean between two terms a & b is calculated by; 

  3rd mean  =(a+3d)
  4th mean  =(a+4d)

  ……………………………………………………………
  ……………………………………………………………

  ……………………………………………………………

   nth mean =a+nd

Sum of AP
Let the A.P. be a, a+d, a+2d…………l-2d, l-d, l(l= last term)

The sum of series;                 S_n= a+(a+d) +(a+2d)+…+ (l-2d) +  (l-d)  + l….(i)

This can also be written as; S_n= l+  (l-d)   + (l-2d) +…+(a+2d)+ (a+d)+ a……(ii)

Adding equation (i) & (ii),

 2S_n=(a+l)+(a+l)+(a+l)+…………….. (a+l)+(a+l)+(a+l)

 2S_n=n(a+l)

     which is the required formula to calculate sum of A.P. when first and last term is known.

Also,

tn=a+(n-1)d

If tn term is considered as last term (l).

We get,

which is the required formula to calculate the sum of series of A.P. when first term, common difference and number of terms is known.