Question

2*3/5*9+3*5/9*13+4*7/13*17+5*9/17*21+……+24*47/93*97+25*49/97*101裂项相

Answer 1

问：2*3/5*9+3*5/9*13+4*7/13*17+5*9/17*21+……+24*47/93*97+25*49/97*101裂项相

答：通项为：(n+1)(n+2)/(4n+1)(4n+5)

答：通项为：(n+1)(2n+1)/(4n+1)(4n+5)设该项裂项为：A/(4n+1)-A/(4n+5)+B=(4A+16Bn^2+24Bn+5B)/(4n+1)(4n+5)故，4A+16Bn^2+6Bn+5B=(n+1)(2n+1)16Bn^2+24Bn+5B+4A=2n^2+3n+1有：16B=224B=35B+4A=1解得：A=3/32, B=1/8

答：故通项裂项为：(n+1)(2n+1)/(4n+1)(4n+5)=3/32*[1/(4n+1)-1/(4n+5)]+1/8

答：对于2*3/5*9+3*5/9*13+4*7/13*17+5*9/17*21+……+24*47/93*97+25*49/97*101，n=24，原式=24*1/8+3/32*(1/5-1/9+1/9-1/13+...+1/97-1/101)=3+3/32*(1/5-1/101)=3+3/32*96/505=3+9/505=1524/505

问：还没解答出来吗

答：而且显示已读，1524/505就是最后的结果。

答：通项为：(n+1)(2n+1)/(4n+1)(4n+5)设该项裂项为：A/(4n+1)-A/(4n+5)+B=(4A+16Bn^2+24Bn+5B)/(4n+1)(4n+5)故，4A+16Bn^2+6Bn+5B=(n+1)(2n+1)16Bn^2+24Bn+5B+4A=2n^2+3n+1有：16B=224B=35B+4A=1解得：A=3/32, B=1/8

答：故通项裂项为：(n+1)(2n+1)/(4n+1)(4n+5)=3/32*[1/(4n+1)-1/(4n+5)]+1/8

答：对于2*3/5*9+3*5/9*13+4*7/13*17+5*9/17*21+……+24*47/93*97+25*49/97*101，n=24，原式=24*1/8+3/32*(1/5-1/9+1/9-1/13+...+1/97-1/101)=3+3/32*(1/5-1/101)=3+3/32*96/505=3+9/505=1524/505