114. Write 271 as the Sum of Positive Real Numbers
Write 271 as the sum of positive real numbers so as to maximize their product.
Submitted by tartle · Added 4 December 2010
Solution:
Split 271 into 100 equal terms:
271 = 2.71 + 2.71 + … + 2.71 (100 times)
This yields the maximal product, (2.71)^100.
Comments (6)
2+3+4+5+6+7+8....
note:0 is not applied cuz then product becomes 0. 1 does not change the product...am i right?
135.5+135.5
3+2+2+2+2+2.......
Product = 3 x 134th power of 2
3+2+2+2+2+2.......
Product = 3 x 134th power of 2
It's 2.71 repeated 100 times.
This appears to be correct, but needs a little explanation.
(n+1)*(n-1) = n^2-1, which is less than n^2. This shows that uniform values adding to a given sum make the largest product. Therefore, using a for 271, the product y of x uniform values can be written as:
y = (a/x)^x = e^[x*(lna-lnx)]
The derivative is:
dy/dx = [(a/x)^x]* = [(a/x)^x]*
At the maximum product, the derivative equals zero, so that:
ln(a/x) = 1;
a/x = e;
x = a/e = 271/2.718... = 100 to the nearest integer.
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