Proving by Induction: Sequence Increasing & Bounded Above by 3

[seto6]

Homework Statement

Let a₁=1 and a_n+1= \sqrt{1+2a_{n}}
where n is a natural number . Use the principle of mathematical induction to prove that the sequence {a_n} is increasing and bounded above by 3. conclude that the sequence converges and find its limit

2. The attempt at a solution

i can see that it converges to 1, and for induction i think you have to test for n=1 has to be true then n=k has to be true and n=k+1 has to be true but if it was a simple series like the summation of x from x=1 to n, i would be able to do it, i kinda lost. help please
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[LCKurtz]

Homework Statement

Let a₁=1 and a_n+1= \sqrt{1+2a_{n}}
where n is a natural number . Use the principle of mathematical induction to prove that the sequence {a_n} is increasing and bounded above by 3. conclude that the sequence converges and find its limit

2. The attempt at a solution

i can see that it converges to 1

No, it doesn't converge to 1.

and for induction i think you have to test for n=1 has to be true then n=k has to be true and n=k+1 has to be true...

n=1 has to be true? n = k has to be true?? n = k+1 has to be true?

That is a very confused statement. Your first step in solving a problem like this is to correctly and carefully write down what you are assuming for the induction hypothesis and what it is that you are trying to prove.

 

[seto6]

isnt the limit as n-> infinity for an and an+1 the same then

let lim n->infinity an=L so an+1=l

so you got L=(1+2L)^.5 solving for L you get (L^2 -2L -1) =(L-1)^2 there L=1and i have to prove that it is increasing and its bounded above by 3

i don't know how to prove this using induction

 

[radou]

As LCKurzt suggested, do it step by step. First you have to prove it holds for n = 1. Then assume it holds for n = k. Then inquire what happens for n = k + 1 (using the assumption it holds for n = k).

 

[seto6]

but the problem is its a sequence, if it was a series it would of been much easier to test for n=1, n=k and test for n=k+1

 

[LCKurtz]

isnt the limit as n-> infinity for an and an+1 the same then

let lim n->infinity an=L so an+1=l

so you got L=(1+2L)^.5 solving for L you get (L^2 -2L -1) =(L-1)^2 there L=1

That expression doesn't give (L-1)². Also, until you have shown the sequence has a limit you can't conclude any answer gotten with that method is correct.

and i have to prove that it is increasing and its bounded above by 3

i don't know how to prove this using induction

but the problem is its a sequence, if it was a series it would of been much easier to test for n=1, n=k and test for n=k+1

Induction problems don't have to be sequences or series either one. They just have to be statements about the positive integers. You actually have two problems, to show the sequence is increasing and to show it is bounded above by 3. Why don't you try the bounded by 3 part first:

1. Check it is true for n = 1 (trivial)
2.Write down the induction hypotheses.
3.Write down what you are trying to prove.

Then see what happens...