Prove that the sequence is decreasing monotonically and that it is bounded from below?

The sequence is {a_n}. 

a_1 = 2a_(n+1) = 1/2(x_n+2/x_n)

When finding the first few terms it is seen that the sequence is decreasing, but how do I show that and how do I show that it is bounded below? 

a_(n+1) should be less than or equal to a_n.Thanks in advance.

1 Answer

  • Zac Z
    Lv 7
    2 months ago

    I guess there's a line break missing. Shouldn't it be:

    a_1 = 2

    a_(n+1) = 1/....

    That would make more sense.

    But I'm also wondering about the x_n. There's no definition given, how could you possibly have calculated any terms?

    I'd be good to check whether you've given us all information.

    Also, please use brackets or spaces in your formulas to avoid ambiguities:

    I guess that 1/2(x_... should be understood as ½ (x_... but I'm not entirely sure whether x_n+2/x_n should be "x_(n+2) / x_n" or "x_n + 2/x_n".

    These distinctions are essential.

    • Natalie2 months agoReport

      I'm so sorry. The x_n should be a_n. 
      This was given:
      a_1 = 2
      a_(n+1) = (1/2)[ a_n + 2 / (a_n) ] Again I am very sorry for the confusion.

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