Induction

[Trumbone]

We are given that

a[sub:1ovivgq6]1[/sub:1ovivgq6] = 1

and a[sub:1ovivgq6]n+1[/sub:1ovivgq6] = a[sub:1ovivgq6]n[/sub:1ovivgq6]/(a[sub:1ovivgq6]n[/sub:1ovivgq6] + 1) for n ? 1.
The task is to conjecture a formula for a[sub:1ovivgq6]n[/sub:1ovivgq6] and then prove it using induction.

This is what I have so far,

I found that a[sub:1ovivgq6]n[/sub:1ovivgq6] = 1/n for n ? 1. (Conjecture)

My basis is n = 1......we know that a[sub:1ovivgq6]1[/sub:1ovivgq6] = 1 and then using the conjecture a[sub:1ovivgq6]1[/sub:1ovivgq6] = 1/1 = 1.

Induction hypothesis: Assume this is true when n = k.

Induction Step: n = k+1
If k=1 then a[sub:1ovivgq6]k+1[/sub:1ovivgq6] = a[sub:1ovivgq6]2[/sub:1ovivgq6] = 1/(1+1) = 1/2. Therefore a[sub:1ovivgq6]k+1[/sub:1ovivgq6] = 1/(k+l) for k=1.

I am not sure where to go from here. Does anyone have any ideas? Thanks

 

[daon]

You showed a_2.

Now assuming a_k=1/k, we get a_(k+1) = a_k/(a_k+1) = (1/k)/(1+1/k) = (1/k)/[(k+1)/k] = 1/(k+1).

 

[Trumbone]

Thank you very much