## 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 |

*↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓*

## Winner draw:

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

## Comments 2

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

Posted using Partiko Android

thanks @quantumdeveloper :)