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Prove sequence is increasing

  • Thread starter SMA_01
  • Start date
  • #1
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Homework Statement



Let b1=1 and bn=1+[1/(1+bn-1)] for all n≥2. Note that bn≥1 for all n in N (set of all positive integers).



The Attempt at a Solution



Prove that (b2k-1)k in N

By definition, a sequence (an) is increasing if an≤an+1 for all n in N.

SO, for this problem, must prove b2n-1≤b2n for all n.

Proceed by induction:

Start with n=1.
Then,
b1=1 and b2=3/2, so
b1≤b2.

Assume inductively that b2n-1≤b2n, prove b2n≤b2n+1

Am I doing this correctly? I want to know before I continue.

Thanks.
 
Last edited:

Answers and Replies

  • #2
STEMucator
Homework Helper
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Yes, induction is indeed appropriate here.

Hmm, you could have also treated your sequences like functions and used the first derivative test :)
 
  • #3
218
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Edit: I made a mistake, it's prove b2n-1≤b2n+1
 

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