Question

()/3＝4/1＝16/()＝()除以12

Answer 1

证：n=1时，3/(12×22)=3/41-1/(1+1)2=1-1/4=3/43/(12×22)=1-1/(1+1)2，等式成立。假设当n=k(k∈N+，k≥1)时，等式成立，即3/(12×22)+5/(22×32)+7/(32×42)++(2k+1)/[k2(k+1)2]=1-1/(k+1)2则当n=k+1时，3/(12×22)+5/(22×32)+7/(32×42)++(2k+1)/[k2(k+1)2]+[2(k+1)+1]/[(k+1)2(k+1+1)2]=1-1/(k+1)2+(2k+3)/[(k+1)2(k+2)2]=1-[(k+2)2-(2k+3)]/[(k+1)2(k+2)2]=1-(k2+4k+4-2k-3)/[(k+1)2(k+2)2]=1-(k2+2k+1)/[(k+1)2(k+2)2]=1-(k+1)2/[(k+1)2(k+1+1)2]=1-1/[(k+1)+1]2，等式同样成立。k为任意正整数，因此等式对于任意正整数n恒成立。3/(12×22)+5/(22×32)+7/(32×42)++(2n+1)/[n2(n+1)2]=1-1/(n+1)2