Is a(n) bounded by a(1)*c^(n-1)?

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Homework Help Overview

The discussion revolves around a sequence defined by a recurrence relation, specifically examining whether the terms of the sequence are bounded by a specific expression involving a constant and the first term of the sequence. The subject area includes sequences and mathematical induction.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the use of mathematical induction to prove the boundedness of the sequence. Questions arise regarding the establishment of a base case and the implications of the recurrence relation.

Discussion Status

There is an ongoing exploration of the proof by induction approach, with some participants questioning the validity of the base case and others suggesting that it may be trivially true. Multiple interpretations of the base case are being discussed.

Contextual Notes

Participants note the challenge of having two variables in the context of the proof, which may complicate the establishment of a clear base case.

transgalactic
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i am given a real and positive number a1 a2 a3 ...
which goes by a(n)<=c*a(n-1) for every n=>2 for a certain given number c>0 .

prove that
a(n)<=a(1)*c^(n-1)

??

a(n) is the n'th number of the series
 
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Use a proof by induction. It should be fairly straight forward.
 
i don't have a base case here

and there are two variables

??
 
Use induction of n. Your base case would be n=2.
 
for n=2
i get
a(2)=c*a(1)

this base case doesn't prove anything

??
 
The base case is trivially true, since it is the same condition for the members of the sequence.
 

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