(1) Let bn=an−211. Thus, we have an=bn1+21.
Since (1−an)an+1=41, then:
(1−(bn1+21))(bn+11+21)=41.
This simplifies to:
bn+1−bn=−2.
Given that b1=41−211=−4,
we can deduce that {bn} is an arithmetic sequence with a first term of −4 and a common difference of −2. Hence, the sequence {an−211} is an arithmetic sequence.
(2) Knowing from (1) that bn=−4−2(n−1)=−2n−2,
we get an=bn1+21=−2n+21+21=2(n+1)n. Therefore:
anan+1=2(n+2)n+1⋅n2(n+1)=n(n+2)(n+1)2=1+n(n+2)1=1+21(n1−n+21).
Summing this from 1 to n, we find:
a1a2+a2a3+…+anan+1=n+21(1−31+21−41+…+n1−n+21).
The series telescopes to:
n+21(1+21−n+11−n+21),
which is less than n+43. Hence we have \boxed{n+ \frac {1}{2}\left(1+ \frac {1}{2} - \frac {1}{n+1} - \frac {1}{n+2}\right) < n+ \frac {3}{4}}.