How to find out the recurrence relation "Un" with the following information

[Astenaskovitch]

U0 = 1
Un+1 = Un * √((Un^2)+1)

 

[Deleted member 4993]

U0 = 1
Un+1 = Un * √((Un^2)+1)

What is the definition of a recurrence relation? Can you provide an example (of recurrence relation) from your textbook (or Google)?

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.

 

[pka]

U0 = 1
Un+1 = Un * √((Un^2)+1)

Can we assume that it is [imath]\left(U_n\right)\sqrt{\left(\left(U_n\right)^2+1\right)}~?[/imath] Or something else?

 

[Dr.Peterson]

U₀ = 1
U_n+1 = U_n * √((U_n^2)+1)

That is a recurrence relation! What are you supposed to do with it? List the first several terms of the sequence, perhaps? Please state the entire problem as given to you.

 

[HallsofIvy]

I presume you mean "solve the recurrance relation". You are told that U0= 1 and \(\displaystyle U_{n+1}= U_n\sqrt{U_n^2+ 1}\) so \(\displaystyle U_1= 1\sqrt{1^2+ 1}= \sqrt{2}\), \(\displaystyle U_2= (\sqrt{2})(\sqrt{(\sqrt{2})^2+ 1})= \sqrt{2}\sqrt{3}= \sqrt{6}\), \(\displaystyle U_3= (\sqrt{6})(\sqrt{6+1})= \sqrt{42}\), etc.

Do you see a pattern?