44a << ………15分
综上，a 的取值范围为)4
25
,(-∞ ． ………16分
20. （1）设数列{}n a 的公差为d ，由题设可得2(1)1(14)d d +=?+．
解得 d=0（舍）或d=2，所以21n a n =-． ………2分 由2
43n n S
S +=+，可得214(1)n n S S ++=+ ………4分
又因为11b =，22b =，所以112S +=，214S +=． 当n 为奇数时，12n
n S +=； 当n 为偶数时，12n n S +=．
所以12,n
n S n *
+=∈N ……6分 当2n ≥时，1
12n n n n b S S --=-=，
所以1
2
,n n b n -*=∈N ． ……8分
（2）因为 1
1(21)(2
1)(21)2(21)n n n c n n n --=--=-?--，
则0
1
2
2
12123252(23)2
(21)2n n n T n n n --=?+?+?++-+-- ． 设0
1
2
2
1123252(23)2(21)2n n n M n n --=?+?+?++-+-
则1
2
1
21232(23)2
(21)2n n n M n n -=?+?++-+-
两式相减，得2
3
1222(21)2(23)23n
n
n
n M n n -=++++--=--+ 所以(23)23n
n M n =--．
所以2
(23)23n n T n n =--- ……12分 令 2
(3)(23)2n
n n e n T n n n =+-=-， 由1n n e e +< ，得1(23)2(1)(21)2n n n n n n +-<+- 即(23)2(1)(21)n n n n -<+-，解得对任意*n ∈N 成立，
即数列{}n e 为单调递增数列． …14分 当n 为奇数时，12e λ-=-≤，所以2λ≥； 当n 为偶数时，28e λ=≤，
所以28λ≤≤． ……16分
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