Prove using induction that x_n = (2+2*sqrt(2))*x_{n-1}*(1-x_

[cdx]

If x[sub:3pcyk6st]0[/sub:3pcyk6st] = (sqrt(2) / 2), prove using induction that x[sub:3pcyk6st]n[/sub:3pcyk6st] = (2 + 2*sqrt(2)) * x[sub:3pcyk6st](n-1)[/sub:3pcyk6st] * (1 - x[sub:3pcyk6st](x-1)[/sub:3pcyk6st]) will remain constant for every value of n.

I do not completely understand proof by induction but I have tested a few base cases (n=1, n=2, etc.) and have found that the answer remains constant at (sqrt(2) / 2).

Where do I go from here?

 

[Deleted member 4993]

Re: Prove using induction

If x[sub:35d2dcb9]0[/sub:35d2dcb9] = (sqrt(2) / 2), prove using induction that x[sub:35d2dcb9]n[/sub:35d2dcb9] = (2 + 2*sqrt(2)) * x[sub:35d2dcb9](n-1)[/sub:35d2dcb9] * (1 - x[sub:35d2dcb9](x-1)[/sub:35d2dcb9]) will remain constant for every value of n.

I do not completely understand proof by induction but I have tested a few base cases (n=1, n=2, etc.) and have found that the answer remains constant at (sqrt(2) / 2).

Where do I go from here?

Go to google and type in:

proof by induction mathematics

You'll find many sites with interactive examples.

If you are still stuck, write back showing your work and indicating exactly where you are stuck.

 

[cdx]

Re: Prove using induction

I have browsed Google for proof by induction articles/examples but I am still confused as to how to do this.

I believe I am attempting to solve for the n+1 case.

If I let n=1,
x[sub:3hru9y3e]1[/sub:3hru9y3e] = (2 + 2*sqrt(2)) * x[sub:3hru9y3e](1-1)[/sub:3hru9y3e] * (1 - x[sub:3hru9y3e](1-1)[/sub:3hru9y3e])
x[sub:3hru9y3e]1[/sub:3hru9y3e] = (2 + 2*sqrt(2)) * x[sub:3hru9y3e]0[/sub:3hru9y3e] * (1 - x[sub:3hru9y3e]0[/sub:3hru9y3e])
x[sub:3hru9y3e]1[/sub:3hru9y3e] = (2 + 2*sqrt(2)) * (sqrt(2) / 2) * (1 - (sqrt(2) / 2))
...
x[sub:3hru9y3e]1[/sub:3hru9y3e] = (sqrt(2) / 2)

The next step in these articles/examples states to let n = k. This is just a substitution and I understand this part, however, where I go from there is what confuses me. Many of the articles/examples cite the n(n+1)/2 example but that is a summation and doesn't seem to apply to my problem. Any further help would be appreciated.

 

[cdx]

Re: Prove using induction

If I substitute (n+1) for n in my equation then,
x[sub:18zu8bdj](n + 1)[/sub:18zu8bdj] = (2 + 2*sqrt(2)) * (x[sub:18zu8bdj]((n + 1) - 1)[/sub:18zu8bdj] * (1 - x[sub:18zu8bdj]((n + 1) - 1)[/sub:18zu8bdj])

becomes

x[sub:18zu8bdj](n + 1)[/sub:18zu8bdj] = (2 + 2*sqrt(2)) * (x[sub:18zu8bdj]n[/sub:18zu8bdj]) * (1 - x[sub:18zu8bdj]n[/sub:18zu8bdj])

Now, do I substitute my original equation of x[sub:18zu8bdj]n[/sub:18zu8bdj] = (2 + 2*sqrt(2)) * x[sub:18zu8bdj](n-1)[/sub:18zu8bdj] * (1 - x[sub:18zu8bdj](n-1)[/sub:18zu8bdj]) into this new equation for x[sub:18zu8bdj]n[/sub:18zu8bdj]?

 

[Deleted member 4993]

Re: Prove using induction

If x[sub:13aqnbv9]0[/sub:13aqnbv9] = (sqrt(2) / 2), prove using induction that
Assume
x[sub:13aqnbv9]n[/sub:13aqnbv9] = (sqrt(2) / 2)

Then
x[sub:13aqnbv9]n+1[/sub:13aqnbv9]

= (2 + 2*sqrt(2)) * x[sub:13aqnbv9](n)[/sub:13aqnbv9] * (1 - x[sub:13aqnbv9](n)[/sub:13aqnbv9])

= (2 + 2*sqrt(2)) * (sqrt(2) / 2) * (1 - (sqrt(2) / 2))

simplify and show that the above reduces to (sqrt(2) / 2)



will remain constant for every value of n.

I do not completely understand proof by induction but I have tested a few base cases (n=1, n=2, etc.) and have found that the answer remains constant at (sqrt(2) / 2).

Where do I go from here?

 

[cdx]

Re: Prove using induction

I accidentally included an 2* in my equation. The original equation should read, x[sub:15890pf6]n[/sub:15890pf6] = (2 + sqrt(2)) * (x[sub:15890pf6](n-1)[/sub:15890pf6]) * (1 - x[sub:15890pf6](n-1)[/sub:15890pf6]).


Correct me if I am wrong, but based upon what you've written this is how I should proceed with my problem...

x[sub:15890pf6](n+1)[/sub:15890pf6] = (2 + sqrt(2)) * (x[sub:15890pf6]n[/sub:15890pf6]) * (1 - (x[sub:15890pf6]n[/sub:15890pf6]))

where x[sub:15890pf6]n[/sub:15890pf6] is assumed to equal (sqrt(2)/2), then...

x[sub:15890pf6](n+1)[/sub:15890pf6] = (2 + sqrt(2)) * (sqrt(2)/2) * (1 - (sqrt(2)/2))
x[sub:15890pf6](n+1)[/sub:15890pf6] = [2 + (2*sqrt(2))/2] * [1 - sqrt(2)/2]
x[sub:15890pf6](n+1)[/sub:15890pf6] = [(4*sqrt(2)) - 4 + 4 - (2*sqrt(2))] / 4
x[sub:15890pf6](n+1)[/sub:15890pf6] = (2*sqrt(2)) / 4
x[sub:15890pf6](n+1)[/sub:15890pf6] = sqrt(2) / 2

And that is considered a proof by induction?