Increasing or Decreasing Sequence

In summary, the conversation is about a sequence that is defined recursively and the speaker is trying to prove whether it is increasing or decreasing. They provide some terms of the sequence and discuss their observations. They also mention a possible problem with the sequence and ask for advice on how to show that it is convergent.
  • #1
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A sequence (an) is defined recursively by a1 =1 and an+1 = 1/ 2+an 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.
 
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  • #2
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
 
  • #3
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?
 
  • #4
I suspect that your OP is not very well formatted now. Is the sequence
an+1 = (1/2)+an (which is what you wrote using proper order of operations)

or is it

an+1=1/(2+an)
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
 
  • #5
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 :)
 

1. What is an increasing sequence?

An increasing sequence is a series of numbers in which each subsequent number is larger than the previous one.

2. How do I determine if a sequence is increasing?

To determine if a sequence is increasing, look at the difference between each term in the sequence. If the difference between each term is positive, then the sequence is increasing.

3. What is a decreasing sequence?

A decreasing sequence is a series of numbers in which each subsequent number is smaller than the previous one.

4. How can I tell if a sequence is decreasing?

To determine if a sequence is decreasing, look at the difference between each term in the sequence. If the difference between each term is negative, then the sequence is decreasing.

5. How can I find the pattern in an increasing or decreasing sequence?

To find the pattern in an increasing or decreasing sequence, look at the difference between each term in the sequence. The pattern will be the same for each difference, either increasing or decreasing.

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