#0 is like a Practice Problem, the rest are actually a bit harder

(0) Take any number. Now take the square root and add 2. Do this repeatedly. What happens (no matter what you number you start with), and why?

(1) What is the value for n that gives the highest result for the nth root of n?

(2) If you pick numbers randomly from 0 to 1 (inclusive), how many picks would you expect [on average] it would take you for the total sum of the numbers you've drawn to reach or pass 1?

(3) Given the defining property of the Golden Ratio:

p^{2} = p + 1

Prove this formula:

p^{n} = F(n)*p + F(n-1)

and this one:

p^{n} = p^{n-1} + p^{n-2}

Note that p has the quadratic equation:

p^{2} - p - 1 = 0

thus the values are (1+√5)/2 and (1-√5)/2, and both have the properties of p

F(n) = Fibonacci number

Fibonaccis: add the previous two: 1,1,2,3,5,8,13,21,34,55,89,etc

ie, F(7) = 13

also, negative Fibonaccis hold up to the rule that the previous two add to the next one, so a full box might be:

## Question

## unreality 1

#0 is like a Practice Problem, the rest are actually a bit harder

(0) Take any number. Now take the square root and add 2. Do this repeatedly. What happens (no matter what you number you start with), and why?

(1) What is the value for

nthat gives the highest result for thenth root ofn?(2) If you pick numbers randomly from 0 to 1 (inclusive), how many picks would you expect [on average] it would take you for the total sum of the numbers you've drawn to reach or pass 1?

(3) Given the defining property of the Golden Ratio:

p

^{2}= p + 1Prove this formula:

p

^{n}= F(n)*p + F(n-1)and this one:

p

^{n}= p^{n-1}+ p^{n-2}Note that p has the quadratic equation:

p

^{2}- p - 1 = 0thus the values are (1+√5)/2 and (1-√5)/2, and both have the properties of p

F(n) = Fibonacci number

Fibonaccis: add the previous two: 1,1,2,3,5,8,13,21,34,55,89,etc

ie, F(7) = 13

also, negative Fibonaccis hold up to the rule that the previous two add to the next one, so a full box might be:

(4) extra credit for number 3: Can you create a formula based on p & n which calculates F(n) without using any other F(x) in the equation?

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