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# help

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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.

Feb 20, 2020

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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.

Feb 21, 2020