Math Contest #20 Results and Solution


Solution

The problem of this contest was to find the limit of a sequence:
Untitled.png

  1. Does this series converge?
    Yes: This sequence returns the average of x and 2/x in each iteration. And since x will always stay bigger than one this will slowly reduce the difference between the two until an equillibrium is reached.
  2. What is the limit?
    For x → ∞ the difference between two consecutive members of the sequence will go to zero, so for x → ∞:
    x(n+1) = x(n) = ½(x(n) + 2/x(n))
    2x(n) = x(n) + 2/x(n)
    2x(n)² = x(n)² + 2
    x(n)² = 2

    x = √2
  3. Bonus: Construct a similar sequence for another root
    This can be done by reversing above steps:
    x = √a

    x(n)² = a
    2x(n)² = x(n)² + a
    2x(n) = x(n) + a/x(n)
    x(n) = ½(x(n) + a/x(n))

    x(n+1) = ½(x(n) + a/x(n)) is a sequence that converges to √a.

↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓

List of participants with their entries:

Name solutions found comment
@crokkon all correct +bonus
@kaeserotor all correct +bonus

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Winner draw:

2 SBI for 2 person → you both won 1 SBI and 10 STEM each!
↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓

The next contest starts in 2 days. Don't miss it!


Comments 2


Thanks a lot, haven't dealt with sequences for a while, so this one was really entertaining!

Posted using Partiko Android

31.10.2019 20:54
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03.11.2019 13:50
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