Math Contest #20 Results and Solution
Solution
The problem of this contest was to find the limit of a sequence:
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.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
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!
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The next contest starts in 2 days. Don't miss it!
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0.000
Thanks a lot, haven't dealt with sequences for a while, so this one was really entertaining!
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