Math Contest #20 Results and Solution

Solution

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

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

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

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!