Question

Calculați:[1-1/2]•[1-1/3]•[1-1/4]•...•[1-1/2018]=

Answer 1

Varianta fara parte intreaga:

$\prod\limits_{k=1}^{2017} \Big(1- \dfrac{1}{k+1}\Big) = \\ \\ = \prod\limits_{k=1}^{2017} \Big(\dfrac{k+1-1}{k+1} \Big) = \prod\limits_{k=1}^{2017} \dfrac{k}{k+1} = \\ \\ =\dfrac{\prod\limits_{k=1}^{2017} (k)}{\prod\limits_{k=1}^{2017} (k+1) } =\dfrac{\prod\limits_{k=1}^{2017} (k)}{\prod\limits_{k=2}^{2017} (k) \cdot 2018} = \\ \\ =\dfrac{1\cdot \prod\limits_{k=2}^{2017} (k)}{\prod\limits_{k=2}^{2017} (k) \cdot 2018} = \dfrac{1}{2018}$

Varianta cu parte intreaga:
[1-1/2] = [1-0,5] = [0,95] = 0

=> [1-1/2]•[1-1/3]•[1-1/4]•...•[1-1/2018]=
= 0•[1-1/3]•[1-1/4]•...•[1-1/2018]= 0