Two Progressions
5 distinct positive reals form an arithmetic progression. The 1st, 2nd and 5th term form a geometric progression. If the product of these 5 numbers is , what is the product of the 3 terms of the geometric progression?
Details and assumptions
The phase "form an arithmetic progression" means that the values are consecutive terms of an arithmetic progression. Similarly, "form a geometric progression" means that the values are consecutive terms of a geometric progression.
The answer is 8.
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1 solution
let the sequence is ( a , a + d , a + 2d , a + 3d , a+ 4d )
since the first and second and fifth term form a geometric sequence
so (a + 4d ) / ( a + d ) = ( a + d ) / a
so d = 2a
so the sequence becomes ( a , 3a , 5a , 7a , 9a )
The product of the 5 numbers is 1120 / 9
so a X 3a X 5a X 7a X 9a = 1120 / 9
a = 2 / 3
so the sequence is ( 2 / 3 , 2 , 10 / 3 , 14 / 3 , 6 )
the product of the geometric sequence is 2 / 3 X 2 X 6 = 8