Question

1/2+1/5+1/9+1/14+1/20+1/27+……+1/20099=?简算

2=2 ,5=2+3,9=2+3+4,14=2+3+4+5……

Answer 1

1/2=1/(1+2-1)
1/5=1/(1+2+3-1)
1/9=1/(1+2+3+4-1)
1/14=1/(1+2+3+4+5-1)
…………
1/20099=1/(1+2+3+4+...+201-1)
第k项=1/[1+2+...(n+1)-1]
=1/[(n+1)(n+2)/2 -1]
=1/[(n²+3n+2-2)/2]
=1/[(n²+3n)/2]
=2/[n(n+3)]
=(2/3)[1/n -1/(n+3)]
1/2+1/5+...+1/20099
=(2/3)(1/1-1/4+1/2-1/5+...+1/200-1/203)
=(2/3)[(1/1+1/2+...+1/200)-(1/4+1/5+...+1/203)]
=(2/3)(1+1/2+1/3 -1/201-1/202-1/203)
=(2/3)[(267×202×203+201×100×203-201×202]/(201×202×203)
=4996100/4121103