A new updating law:

y[1]  =  125;
y[n_]  :=  
      125  +  (1  +   .06*Cos[((2  Pi n)/28)]) ( .6) 
          y[n - 1]

ListPlot[
      Table[{n,  y[n]},  {n, 1, 50}]]

ListPlot[
      Table[{n,  y[n]},  {n, 30, 100}]]

ListPlot[Table[{n,  y[n]}, {n, 100, 200}]]

I tried to fit the data with a curve of the form
y(t)  = A ( 1 + B  Cos[((2 π (t - C))/28)]).  
Notice  I have three parameters to choose.  My first guess would be  
A = 312.5,   B = .1,  C = 0.  
 I had to adjust the parameters a bit to fit the curve.
I found that 
A = 313.6,   B = .083,  C =  1.3. 
worked pretty well.  But these parameters were chosen just by guesswork and observation, not by any systematic conceptual method.

plot1  =  
      ListPlot[Table[{n,  y[n]},  {n, 100, 200}],  
        DisplayFunction  ->  Identity]
f[t_]  :=  
      313.6 (1  +   .083  Cos[((2 Pi (t - 1.3))/28)])
plot2  =  
      Plot[f[t],  {t, 100, 200},  
        DisplayFunction  ->  Identity]
Show[{plot1,  plot2},  
      DisplayFunction  -> $DisplayFunction]

plot1  =  
      ListPlot[Table[{n,  y[n]},  {n, 120, 148}],  
        DisplayFunction  ->  Identity]
plot2  =  
      Plot[f[t],  {t, 120, 148},  
        DisplayFunction  ->  Identity]
Show[{plot1,  plot2},  
      DisplayFunction  -> $DisplayFunction]

Max[ Table[Abs[y[n] - f[n]],  {n,  100,  200}]]

Max[Table[y[n],  {n, 1, 250}]]