Is the Sequence (b2n-1)k in N Increasing?

[SMA_01]

Homework Statement

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

The Attempt at a Solution

Prove that (b_2k-1)_{k in N}

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

SO, for this problem, must prove b_2n-1≤b_2n for all n.

Proceed by induction:

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

Assume inductively that b_2n-1≤b_2n, prove b_2n≤b_2n+1

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

Thanks.

 

[STEMucator]

Yes, induction is indeed appropriate here.

Hmm, you could have also treated your sequences like functions and used the first derivative test :)

 

[SMA_01]

Edit: I made a mistake, it's prove b_2n-1≤b_2n+1