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

skylsn 2022-06-26 Answered
The sequence is defined by recurrence: x 1 = 1, x 2 = 2
x n + 1 = 1 2 ( x n + x n 1 )
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Answers (1)

Harold Cantrell
Answered 2022-06-27 Author has 21 answers
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.)

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