Arithmetic Progression

How to calculate the common difference when the first term and the nth term is know. 

Question:

The 28th term of an AP is -5. Find the common difference if its first term is 31.

Solution

T1 = a ( First term of the sequence)

a = 31

T28 = -5 (28th term of the sequence)

d = ? (common difference)

Recall the nth term formula of Ap

i.e  Tn = a + (n-1)d

       Substitute 28 for n to get the following equation

       T28 = a + (28-1)d

         T28     = a + 27d

       Substitute T28 for -5 and 31 for a  in the equation above 

          -5     = 31 + 27d

        Subtract 31 from both sides of the equation

        -5 -31 = 27d

         -36 = 27d

       Divide through by 27

       -36/27  =  27d/27

       d = -36/27

       d = -4/3

The common difference is -4/3

CHECK

Substitute the first term and the common difference into the equation below,

  T28     = a + 27d

   T28    = 31 + 27 * -4/3

Divide 27 by 3 to get 9

    T28   = 31 - 9 * 4

    T28   = 31 - 36

    T28  = - 5