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问题: 必修五数学3

尽量详细!!!

解答:

(I)等差数列--->√f(x)=√x+√3
--->√S(n+1)=√f(Sn)=√Sn+√3
--->{√S(n)}是等差数列,√S1=√3=d--->√Sn=n√3
--->Sn=3n²,S(n+1)=3(n+1)²
--->a(n+1)=S(n+1)-Sn=3(2n+1)

(II) an=3(2n-1),a(n+1)-an=6--->bn=1/[ana(n+1)]
--->Tn= b1+b2+...+bn
   = 1/(a1a2)+1/(a2a3)+...+1/[ana(n+1)]
   =(1/6){(a2-a1)/(a1a2)+(a3-a2)/(a2a3)+...+[a(n+1)-an]/[ana(n+1)]}
   =(1/6)[(1/a1-1/a2)+(1/a2-1/a3)+...+1/an-1/a(n+1)]
   =(1/6)[1/a1-1/a(n+1)]
   =(1/6)[1/3-1/3(2n+1)]
   =(1/18)[2n/(2n+1)]......<1/18
   =(1/9)[1/(2+1/n)]...... ≥(1/9)[1/(2+1/1)]=1/27