A-sequence-U-n-is-defined-reculsively-as-U-o-1-2-and-U-n-1-2-1-U-n-for-n-N-a-Show-by-mathematical-induction-that-all-terms-in-the-sequence-are-positive-b-Given-that-t

Question Number 87308 by Rio Michael last updated on 03/Apr/20

A sequence (U_{n}) is defined reculsively as

U_{o} = \frac{1}{2} and U_{n + 1} = \frac{2}{1 + U_{n}} for n \in N

a) Show by mathematical induction that all terms in the sequence

are positive .

b) Given that the sequence (U_{n}) is convergent, show that the limit, l, is

a solution to the equation x^{2} + x - 2 = 0 . Hence find l

c) Given that (V_{n}) is a sequence of general term such that

V_{n} = \frac{U_{n} - 1}{U_{n} + 2}, \forall n \in N .

show that (V_{n}) is convergent and determine its limit .

hence deduce the convergence of the sequence (U_{n}) .

P l e a s e r e c o m m e n d m e t e x t b o o k s f o r t h i s t o p i c e v e n y o u t u b e v i d s

p l e a s e