Show that sequence is increasing: a_1 = 1, a_{n+1} = 3 - 1/(a_n) for n >= 1

[KamCh]

Hello, can you please help with the question of showing that sequence is increasing using induction?

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Consider the sequence defined recursively by the relations

. . . . .[math]a_1\, =\, 1,\quad a_{n+1}\, =\, 3\, -\, \frac{1}{a_n},\, \mbox{for}\, n\, \geq\, 1[/math]

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Thank you!

 

[KamCh]

Is this an adequate explaining, I also found limit and showed that sequence bounded by 3 from above. Thank you!

 

[Steven G]

Hello, can you please help with the question of showing that sequence is increasing using induction?

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Consider the sequence defined recursively by the relations

. . . . .[math]a_1\, =\, 1,\quad a_{n+1}\, =\, 3\, -\, \frac{1}{a_n},\, \mbox{for}\, n\, \geq\, 1[/math]

Just show that \(\displaystyle 3-\frac{1}{x}>x\) assuming x>0

Do you have to use induction??

 

[KamCh]

Induction isn't mentioned but I think prof. expects that

 

[Steven G]

Like I said just show that 3−1x>x assuming x>0
This leads to a quadratic equation. Try it!