Insert A.M.’s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.’s.

Let   be n A.M.'s inserted between 7 and 71.

∴  7, are in A.P.
                                 a = 7,
Let d be the common difference
                                       a + (n + 2 - 1)d = 71
     7 + (n + 1)d = 71                                            ...(i)
Now,                                or             
or            a + 5d = 27                      or           7 + 5d = 27
or                 5d = 20                       or                  d = 4                  ...(ii)
From (i) and (ii), we have
                                   or                   n + 1 = 16
or                        n = 15

∴      15 A.M.'s are inserted between 7 and 11.