- #1
rbnphlp
- 54
- 0
Q.Show that any integral power of [tex](\sqrt{2}-1)^k[/tex] can be wriitten as the [tex]\sqrt{N_k}-\sqrt{N_k-1}[/tex] for N a positive integer .
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I tried using induction to solve the problem but when looked at the book's solution they seem to suggest an extra condition that the following equation has to satisfy :
[tex](\sqrt{2}-1)^k=\sqrt{N_k}-\sqrt{N_k-1}[/tex] for N_k a positive integer satisfying [tex]\sqrt{2}\sqrt{N_k}\sqrt{N_k-1} \in \mathbb{Z}[/tex] .
Why is this last condition required to apparently "complete" induction argument..which I don't get why this would be necessary ?
I managed to show the above is true for n+1 , but why would I have to show the above condition is true?
Thanks
.......................................
I tried using induction to solve the problem but when looked at the book's solution they seem to suggest an extra condition that the following equation has to satisfy :
[tex](\sqrt{2}-1)^k=\sqrt{N_k}-\sqrt{N_k-1}[/tex] for N_k a positive integer satisfying [tex]\sqrt{2}\sqrt{N_k}\sqrt{N_k-1} \in \mathbb{Z}[/tex] .
Why is this last condition required to apparently "complete" induction argument..which I don't get why this would be necessary ?
I managed to show the above is true for n+1 , but why would I have to show the above condition is true?
Thanks