DynamicModule[{code = Hold[ f[x_]:> (x^2 - 4)/(x -2)], a= 1/n, b=1},With[{fn = (code = #; ReleaseHold[ # /. RuleDelayed -> SetDelayed]) &},
Manipulate[
Column[{
Labeled[ InputField[ Dynamic[  code, fn], Hold[Expression]], "function", Left], 
Labeled[ InputField[Dynamic[a]], "sequence", Left],
Labeled[InputField[Dynamic[b]],"i=" , Left ], 
Pane[
Dynamic[
 If[ 
( t - b )<11,
TableForm[
 Transpose[{
Range[b, t], 
Table[ N[ a, 4 ], {n, b, t} ],
Table[ N[ f[ a ], 4], {n, b, t}]
}],
TableHeadings->{ None, { "n", "(Subscript[a, n])", " f((Subscript[a, n]))"}}
],
(* else *)
TableForm[
Transpose[{
Range[t-10, t],
Table[ N[ a, 4 ], {n, t-10, t} ],
Table[N[ f[ a ], 4], {n, t-10, t}]
}],
TableHeadings->{ None, { "n", "(Subscript[a, n])", " f((Subscript[a, n]))"}}
]
] (* end If *)
], (* end Dynamic *)
{ 300, 190 }] (* end Pane *)
}], (* end Column *)
{{t, 1, "k"}, 1, 1000, 1, Appearance->"Labeled"},
Initialization:> fn[code]]]]