Column count \ (\ {a_n \} \ ) before \ (n-\) item and as \ (S_n \) , known \ (2S_n-a_n + 1 = n (a_n + 1), \) and \ ( a_2 = 5. \) if \ (m> \ dfrac {S_n } {2 ^ n}, \) is the real number \ (m \) ranges \ (\ underline {\ qquad \ qquad} \) .
Analysis: a title \ [\ begin {cases} & 2S_n-a_n + 1 = n (a_n + 1), \\ & 2S_ {n + 1} -a_ {n + 1} + 1 = (n + 1) ( a_ {n + 1} +1)
, \ end {cases} \] two difference formula and analyzed for available
\ [na_ {n + 1}
-. (n + 1) a_n + 1 = 0 \] on both sides with the other in \ (n-(n-+. 1) \) , available
\ [\ dfrac {a_ {n + 1}} {n + 1} - \ dfrac {1} {n + 1} = \ dfrac {a_n} {n } -. \ dfrac {1}
{n} \] so
\ [\ dfrac {a_n} {
n} - \ dfrac 1n = \ cdots = \ dfrac {a_2} {2} -. \ dfrac 12 = 2 \] thus \ (2N + = A_N. 1, n-\ in \ mathbb {N} ^ \ AST \) , so
\ [S_n = n ^ 2 +
2n, n \ in \ mathbb {N} ^ \ ast. \] followed by Study of the number of columns \ (\ left \ {\ dfrac {S_n} {2 ^ n} \ right \} \) of monotonicity is obtained at its maximum. Because of
\ [\ dfrac {S_ {n + 1}} {2 ^ { n + 1}} - \ dfrac
{S_n} {2 ^ n} = \ dfrac {3-n ^ 2} {2 ^ {n + 1}}, n \ in \ mathbb {N} ^ \ ast \]. Therefore, the number of columns \ (\ left \ {\ dfrac {S_n} {2 ^ n} \ right \} \) in\ (n = 2 \) to obtain the maximum value, so that the required \ (m \) ranges \ ((2, + \ infty) \) .