Question

Arată că 1/2×3+1/3×4+1/4×5+....+1/2015×2016<1/2

Answer 1

1/(2 x 3)=1/2 - 1/3
1/(3 x 4)=1/3 - 1/4
1/(4 x 5)=1/4 - 1/5
..........................
1/(2014 x 2015)=1/2014 - 1/2015
1/(2015 x 2016)=1/2015 - 1/2016
1/2x3 + 1/3x4 + 1/4x5 + .........+1/2014x2015+1/2015x2016 = 1/2 - 1/2016=
=(2016-2)/2x2016=2014/2x2016 =1007/2016

1/2 = 1007/2014 > 1007/2016
prin urmare suma din enunt e mai mica decat 1/2