等差数列(an)的公差为d>0,a1>0,证明1/(a1*a2)+1/(a2*a3)+1/(a3*a4)……+1/(an*an+1)=(1/d)(1/a1-1/an+1)

问题描述:

等差数列(an)的公差为d>0,a1>0,证明1/(a1*a2)+1/(a2*a3)+1/(a3*a4)……+1/(an*an+1)=(1/d)(1/a1-1/an+1)

1/a1*a2=(1/d)*d/(a1a2)=(1/d)(a2-a1)/a1a2=(1/d)[a2/a1a2-a1/a1a2)=(1/d)(1/a1-1/a2)同理1/a2a3=(1/d)(1/a2-1/a3)……1/[ana(n-1)]=(1/d)[1/an-1/a(n+1)]相加,中间正负抵消所以1/(a1*a2)+1/(a2*a3)+1/(a3*a4)……+1...