The fifth,ninth and sixteenth term of an arithmetic sequence are the consecutive term of a geometric sequence. Find the common difference of the arithmetic sequence.
If the arithmetic sequence has an initial term of a and a common difference of d,
the fifth term will be a + 4d,
the ninth term will be a + 8d,
the fifteenth term will be a + 15d.
Since these three terms are the first three terms of a geometric sequence, call the common ratio r.
Therefore, (a + 4d)r = a + 8d and (a + 8d)r = a + 15d.
(a + 4d)r = a + 8d ---> ar + 4dr = a + 8d ---> ar = a + 8d - 4dr
(a + 8d)r = a + 15d ---> ar + 8dr = a + 15d ---> ar = a + 15d - 8dr
Combining these two equations into one equation: a + 8d - 4dr = a + 15d - 8dr
---> 8d - 4dr = 15d - 8dr
---> 4dr = 7d
---> 4r = 7
---> r = 7/4
Since we know that (a + 4d)r = a + 8d ---> (a + 4d)·(7/4) = a + 8d
---> 7a + 28d = 4a + 32d
---> 3a = 4d
---> d = (3/4)·a
This leads to an infinite set of solutions: for example, if a = 4 and d = 3, you get the answers 16, 28, 49
and, if a = 8 and d = 6, you get the answers 32, 56, 98.