Is the system is a geometric progression? I am trying to show that the following system is a geomet

Blericker74 2022-07-04 Answered
Is the system is a geometric progression?
I am trying to show that the following system is a geometric progression i.e. a 2 = a 1 2 , a 3 = a 1 3 etc.
a k 2 = a k 1 a k + 1
a k 1 2 = a k 2 a k
a 2 2 = a 1 a 3
here a k > . . . > a 2 > a 1 is assumed.
Is there sufficient information to conclude that a i = a 1 i in general? I am seeing that I would probably need to assume a 2 = a 1 2 . Could the geometric progression be concluded otherwise?
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Answers (1)

Giovanna Erickson
Answered 2022-07-05 Author has 14 answers
Step 1
a k + 1 a k = a k a k 1 = = a 3 a 2 = a 2 a 1 so { a k } forms a geometric sequence. let the common ratio be r.
Then the sequence is { a 1 , a 1 r , . . . }
if ( a 1 , r ) = ( 1 , 2 ), then a i a 1 i .
if we assume a 2 = a 1 2 r = a 1
Step 2
Then the sequence is { a 1 , a 1 2 , a 1 3 , . . . }
i have excluded the case a 1 = 0, otherwise the sequence is constantly 0.
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