(20 1 1? Shanghai Simulation) Arranges all the items in the series {an} into a numerical table according to the rule that there are three items in the first row, and each row below has one more item than the previous row: Note (1) times (n+1) bn+12-nbn2+bn+1bn = 0, bn > 0,

Let t = bn+ 1bn get t > 0, (n+ 1)t2+t-n=0(6 points).

That is, (t+ 1)[(n+ 1)t-n]=0,

So bn+ 1bn = nn+ 1 (8 points)

So B2B 1 = 12, B3 B2 = 23, …, bnbn? 1=n？ 1n, and bn =1n is obtained by various multiplications;

(2) Let the common ratio of each row in the table above be q, and q > 0. Because 3+4+5+…+ 1 1=63, lines 1 and 9 * * * in the table contain the first 63 items of the series bn, so a66 is in line 10. Q=2。 = 25 and B 10 = 1 10, so Q = 2. Then S (k) = BK (1? qk+2) 1？ q= 1k(2k+2？ 1)k∈N*

(3) When x∈[ 1 1000,100], ∫ 1x? X is the subtraction function, and the minimum value is 100? 1 100,∴ 1k(2k+2？ 1)> 100? 1 100,∴k≥8