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# Proof by induction

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1. .

Prove by induction that .
How do you deal with these inequality types? It seems so trivial but how to give a sound proper proof. It seems that you rearrange it to
.
Then it seems trivial.
2. (Original post by Ano123)
.

Prove by induction that .
As a general rule:

1. Show it works for case n=1 or another easy to evaluate one.

2. Assume it works for n=k.

3. Show that it must then work for n=k+1, normally involves relating to the n=k case.

4. Hence holds by induction
3. (Original post by samb1234)
As a general rule:

1. Show it works for case n=1 or another easy to evaluate one.

2. Assume it works for n=k.

3. Show that it must then work for n=k+1, normally involves relating to the n=k case.

4. Hence holds by induction
What would you make the inductive step?
4. (Original post by Ano123)
What would you make the inductive step?
Showing that the statement holds for given that it holds for .
5. (Original post by Ano123)
What would you make the inductive step?
Assume that Uk>5, then using your rearranged form for Un+1 can easily show that if Uk is greater than 5 than so is Uk+1
6. (Original post by samb1234)
Assume that Uk>5, then using your rearranged form for Un+1 can easily show that if Uk is greater than 5 than so is Uk+1
That's what I've done and demonstrated in the OP, but how can I formally say it. It doesn't seem right saying, it is clearly greater than 5 as long as .
But in this case do you think it's so obvious that it is fine to just do that?
7. (Original post by Ano123)
That's what I've done and demonstrated in the OP, but how can I formally say it. It doesn't seem right saying, it is clearly greater than 5 as long as .
But in this case do you think it's so obvious that it is fine to just do that?
Maybe make a statement that 11/Uk+6 is less than 1 if Uk is greater than 5, and therefore 6 -11/Uk+6 is always greater than 5 if holds for Uk
8. (Original post by samb1234)
Maybe make a statement that 11/Uk+6 is less than 1 if Uk is greater than 5, and therefore 6 -11/Uk+6 is always greater than 5 if holds for Uk
Thank you for the help.

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