Averages/Pythagorean means

Compute all three of the Pythagorean means of the numbers 1..10.

Averages/Pythagorean means
You are encouraged to solve this task according to the task description, using any language you may know.

Show that $A(x_{1},\ldots ,x_{n})\geq G(x_{1},\ldots ,x_{n})\geq H(x_{1},\ldots ,x_{n})$ for this set of positive numbers.

• The most common, Arithmetic mean is the sum of the numbers divided by their count, n:

$A(x_{1},\ldots ,x_{n})={\frac {1}{n}}(x_{1}+\cdots +x_{n})$

• The Geometric mean is the n'th root of the multiplication of all the numbers:

$G(x_{1},\ldots ,x_{n})={\sqrt[{n}]{x_{1}\cdots x_{n}}}$

• The Harmonic mean is n divided by the sum of their reciprocals:

$H(x_{1},\ldots ,x_{n})={\frac {n}{{\frac {1}{x_{1}}}+\cdots +{\frac {1}{x_{n}}}}}$

Lua

<lang lua>function fsum(f, a, ...) return a and f(a) + fsum(f, ...) or 0 end function pymean(t, f, finv) return finv(fsum(f, unpack(t)) / #t) end nums = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

--arithmetic print(pymean(nums, function(n) return n end, function(n) return n end)) --geometric print(pymean(nums, math.log, math.exp)) --harmonic print(pymean(nums, function(n) return 1/n end, function(n) return 1/n end))</lang>

PL/I

<lang PL/I> declare A(10) float static initial (1,2,3,4,5,6,7,8,9,10); n = hbound(A,1);

/* compute the average */ Average = sum(A)/n;

/* Compute the geometric mean: */ Geometric = prod(A)**(1/n);

/* Compute the Harmonic mean: */ Harmonic = n / sum(1/A);

put skip data (Average); put skip data (Geometric); put skip data (Harmonic); </lang>

Python

<lang Python>>>> from operator import mul >>> from functools import reduce >>> def amean(num): return sum(num)/len(num)

>>> def gmean(num): return reduce(mul, num, 1)**(1/len(num))

>>> def hmean(num): return len(num)/sum(1/n for n in num)

>>> numbers = range(1,11) # 1..10 >>> amean(numbers), gmean(numbers), hmean(numbers) (5.5, 4.528728688116765, 3.414171521474055) >>> assert( amean(numbers) >= gmean(numbers) >= hmean(numbers) ) >>> </lang>