Increasing or Decreasing Sequence

[e179285]

A sequence (a_n) is defined recursively by a₁ =1 and a_n+1 = 1/ 2+a_n for all n is greater than 1 or equal 1.

ı'll prove that this sequence is convergent,buy ı cannot decide whethet it is increasing or decreasing .When ı write terms ,some terms increase some terms decrease.

 

[Office_Shredder]

What are the first couple of terms that you got? From the definition in your post it seems like the sequence is increasing to me

 

[e179285]

a1=1/3,a2=7/21,a3=3/7,a4=7/17,a5=17/41 ...

according to my calculation, a1>a2, a3>a2, a5>a4, a3>a4

How can ı decide?

 

[Office_Shredder]

I suspect that your OP is not very well formatted now. Is the sequence
a_n+1 = (1/2)+a_n (which is what you wrote using proper order of operations)

or is it

a_n+1=1/(2+a_n)
which seems to match your most recent post

If it's the latter, then just like you described in your post it's neither increasing nor decreasing

 

[e179285]

My sequence is the second one.If it is neither increasing or decreasing ,how will ı show that it is convergent sequence? ı think there is a problem about sequence.

Thank you for efforts :)