Question

1£®¼ÆËã1+£¨$\frac{1}{2}$+$\frac{1}{3}$+¡­+$\frac{1}{100}$£©+£¨$\frac{2}{3}$+$\frac{2}{4}$+¡­+$\frac{2}{100}$£©+£¨$\frac{3}{4}$+$\frac{3}{5}$+¡­$\frac{3}{100}$£©+¡­+£¨$\frac{98}{99}$+$\frac{98}{100}$£©+$\frac{99}{100}$£®

Answer 1

·ÖÎö ÀûÓüӷ¨µÄ½»»»ÂÉ£¬½«·ÖĸÏàͬµÄ·ÖÊý·ÅÔÚÒ»ÆðÏà¼Ó£¬È»ºóÕÒ³öÆäÖеĹæÂÉ£¬°´ÕÕ¸ß˹ÇóºÍµÄ·½·¨¼ÆËã¼´¿É£®

½â´ð ½â£ºÔ­Ê½=1+$\frac{1}{2}$+£¨$\frac{1}{3}+\frac{2}{3}$£©+£¨$\frac{1}{4}+\frac{2}{4}+\frac{3}{4}$£©+£¨$\frac{1}{5}+\frac{2}{5}$$+\frac{3}{5}+\frac{4}{5}$£©+¡­+$£¨\frac{1}{100}+\frac{2}{100}+¡­+\frac{99}{100}£©$
=1+$\frac{1}{2}+1+\frac{3}{2}+2+¡­+\frac{99}{2}$
=$1+\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+¡­+\frac{99}{2}$
=1+$\frac{1}{2}¡Á\frac{£¨1+99£©¡Á99}{2}$
=1+2475
=2476£®

µãÆÀ ±¾ÌâÖ÷Òª¿¼²éµÄÊÇÓÐÀíÊýµÄ¼Ó·¨£¬ÀûÓÃÓÐÀíÊýµÄ½»»»ÂɺͽáºÏÂÉ£¬½«·ÖĸÏàͬµÄ·ÖÊýÏà¼ÓÊǽâÌâµÄ¹Ø¼ü£®