8.23 Induction Example

8.23 记一道Induction题目

     The sequence of real numbers is such that and

$$a_{n+1}=\left(a_{n}+\frac{1}{a_{n}}\right)^{3}$$

a. Prove by mathematical induction that for all integers .

       [You may use the fact that for .]

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Solution:

a.

               Let ,

                                                       ,

                                                      

                                      So it’s true for .

               Assume it’s true for :

                                                      

                                     
When :

                                                      

                                                      

                                                      

                                                      

                                      So it’s true for

               Hence by induction, it’s true for .