Question

a= 1\1*2 + 1\2*3 + ... + 1\2013*2014
Comparați a cu 1.

Answer 1

$A=\frac{1}{1*2}+\frac{1}{2*3}+...+\frac{1}{2013*2014} \\\\\\ \frac{1}{1*2}=\frac{1}{1}-\frac{1}{2} \\\\ \frac{1}{2*3}=\frac{1}{2}-\frac{1}{3} \\ ................... \\\\ \frac{1}{2013*2014}=\frac{1}{2013}-\frac{1}{2014} \\\\\\A= \frac{1}{1}^{(2014}-\frac{1}{2014} \\\\A= \frac{2014-1}{2014} \\\\ \boxed{\boxed{A=\frac{2013}{2014}<1}}$

Answer 2

1/1*2 poti sa scrii ca 1/1 - 1/2
1/2*3 sa scrii ca 1/2 - 1/3   ,etc   (daca aduci la acc numitor o sa vezi ca o sa-ti dea 1/6, exact ca 1/2*3) si le scrii pe toate ca diferenta
1/2012*2013 = 1/2012 - 1/2013
1/2013*2014 = 1/2013-1/2014
A= 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + .......... 1/2012 - 1/2013 + 1/2013 - 1/2014 
Se se reduc aproape toti termenii, mai ramane doar 1/1 si 1/2014
A= 1 + 1/2014 > 1