4/(1×5)+4/(5×9)+4/(9×13)+...+4/(2013×2017)=
=(5-1)/(1×5)+(9-5)/(5×9)+(13-9)/(9×13)+.....+(2017-2013)/(2013×2017)=
=5/(1×5)-1/(1×5)+9/(5×9)-5/(5×9)+13/(9×13)-9/(9×13)+....+2017/(2013×2017)-
-2013/(2013×2017)=
=1/1-1/5+1/5-1/9+1/9+....-1/2013+1/2013-1/2017=
=1/1-1/2017=
=2017/2017-1/2017=2016/2017
Răspuns:
Explicație pas cu pas:
1 - 1/5 = 5/5 - 1/5 = 4/5
1/5 - 1/9 = 9/5*9 - 5/5*9 = 4/5*9
1/9 - 1/13 = 13/9*13 - 9/9*13 = 4/9*13
...............
1/2013 - 1/2017 = 2017/2013*2017 - 2013/2013*2017 = 4/2013*2017
Suma devine
1 - 1/5 + 1/5 - 1/9 + 1/9 - 1/13 + ...+ 1/2013 - 1/2017 = 1 - 1/2017
= 2017/2017 - 1/2017 = 2016/2017