When three quantities are in A.P., then the middle one is said to be the arithmetic mean of the other two. That is, if a, b and c are in A.P., then b is the arithmetic mean of a and c. Since a, b and c are in A.P., the common difference ought to be constant, we have
b - a
or b + b
or 2b
or b
We should remember that between any two quantities we can insert any number of terms so that the resultant series is in A.P. Now we look at some examples wherein we will insert the required number of terms.
Example
Insert 15 arithmetic means between 4 and 68.
We are given the first term and the last term. The total number of terms including 4 and 68 are therefore 17. Since "a" is given we ought to find "d". The 17th term is given by T_{17} = a + 16d = 4 + 16d. Also the T_{17} term is given to be 68. Therefore,
d
Using the values of "a" and "d" we insert the required terms. The series will be 4, 8, 12, ..., 68.