The sequence is defined by recurrence: x 1 = 1, x 2 = 2 x n + 1 = 1 2 ( x n + x n − 1 )

Question

The sequence is defined by recurrence:       x    1    =  1,       x    2    =  2      x          n      +      1        =      1    2    (      x          n        +      x          n      −      1        )

Harold Cantrell · Accepted Answer

x          n      +      2       is the middle of   [      x    n    ,      x          n      +      1        ], therefore the distance between       x    n   and       x          n      +      1       is divided by 2 at each iteration, namely                    x                  n          +          2                    −              x                  n          +          1                      2    =                    x                  n          +          1                    +              x        n            −      2              x                  n          +          1                      4    =  −      1    2                      x                  n          +          1                    −              x        n              2  (there is a - because   (      x          2      n        ) and   (      x          2      n      +      1        ) are of opposite monotony). Therefore                    x                  n          +          1                    −              x        n              2    =            (      −              1        2            )              n      −      1                          x        2            −              x        1              2    =            (      −      1              )                  n          −          1                            2      n      Thus       x          n      +      1        −      x    n    =            (      −      1              )                  n          −          1                            2              n        −        1            , summing this gives that      x    n    =      x    1    +      ∑          k      =      1              n      −      1        (      x          k      +      1        −      x    k    )  =  1  −      ∑          k      =      1              n      −      1                  (      −              1        2            )              k      −      1        =  1  +            1      −      (      −      1              /            2              )                  n          −          1                            1      +      1              /            2        =  1  +  2            1      −      (      1              /            2              )                  n          −          1                      3  Therefore       lim          n      →      +      ∞            x    n    =      5    3   (you could also solve the characterisic equation of   (      x    n    ) and do the standard calculations.)

The sequence is defined by recurrence: x 1 </msub> = 1 , x 2 </msub>

Answered question

Answer & Explanation

New Questions in Differential Calculus