1/5+1/45+1/117+1/221的简便计算

1/5+1/45+1/117+1/221的简便计算
1/5+1/45+1/117+1/221的简便计算

1/5+1/45+1/117+1/221的简便计算
原式=1／﹙1×5﹚＋1／﹙5×9﹚＋1／﹙9×13﹚＋1／﹙13×17﹚
=¼×﹙1－1／5﹚＋¼×﹙1／5－1／9﹚＋¼×﹙1／9－1／13﹚＋¼×﹙1／13－1／17﹚
=¼×﹙1－1／17﹚
=¼×﹙16／17﹚
=4／17.

裂项
1/4(1-1/5+1/5-1/9+1/9-1/13+1/13-1/17)=4/17

1/5+1/(5×9)+1/(9×13)+1/(13*17)
= (1 - 1/5 + 1/5 - 1/9 + 1/9 - 1/13 + 1/13 - 1/17) /4
= (1 - 1/17) /4
= (17/17 - 1/17) /4
= 16/17/4
= 4/17