Question

DETERMINATI A-2015-A ZECIMALA A NR 1.432( 867)

ARATATI CA 1/ 2X3 +1/3X4 +1/4X5+....+1/2015X2016 <1/2

Answer 1

1)
zecimalele 432 nu sunt in perioada⇒ 3 zecimale nu se repeta
zecimalele 867 sunt in perioada⇒3 zecimale se repeta

2015 - 3= 2012
2012 :3= 670  rest 2 ⇒ zecimala  este a 2a din perioada
⇒ 6 este a 2015 zecimala

2)
(3-2)/2 x 3 + (4-3)/3x4 + (5-4)/4x5+ ... + (2016-2015)/ 2015 x 2016 =
 = 3/2x3 - 2/2x3 + 4/3x4 - 3/3x4 + 5/4x5 - 4/4x5 +..+ 2016/2015x2016 - 2015/2015x2016 =
= 1/2 - 1/3+1/3-1/4+1/4-1/5+...+1/2015 - 1/2016=
= 1/2-1/2016=
= 1008/2016 - 1/2016=
=1007/2016
1/2 = 1008/2016⇒ 1007/2016 < 1/2