In the infinite sequence
a, b, (a + b)/2, (a + 3b)/4, . . .
each term after the second is the arithmetic mean of the two previous terms.

Find the limit of the sequence in terms of real numbers a and b.

It appears to be (a+2b)/3, as follows:

For a=0 and b=1, the sequence from there is:

0.5
0.75
0.625
0.6875
0.65625
0.671875
0.6640625
0.66796875
0.666015625
0.6669921875
0.66650390625
0.666748046875
0.6666259765625
0.66668701171875
0.666656494140625
0.6666717529296875
0.66666412353515625
0.666667938232421875
0.6666660308837890625
0.6666669845581054687
0.6666665077209472656
0.6666667461395263671
0.6666666269302368164
0.6666666865348815917
0.6666666567325592041
0.6666666716337203979
0.666666664183139801
0.6666666679084300994
0.6666666660457849502
0.6666666669771075248
0.6666666665114462375
0.6666666667442768812
0.6666666666278615593
0.6666666666860692203
0.6666666666569653898
0.666666666671517305
0.6666666666642413474
0.6666666666678793262
0.6666666666660603368
0.6666666666669698315
0.6666666666665150842
0.6666666666667424578
0.666666666666628771
0.6666666666666856144

from

10   A=0:B=1
20   for I=1 to 44
30     C=(A+B)/2
40     print C
50     A=B:B=C
60   next

This case appears to approach 2/3, and since the series is linearly related to a and b, the general case would be 2/3 of the way from a to b, or (a+2b)/3.

The first few terms after a and b are

(a+b)/2
(a+3b)/4
(3a+5b)/8
(5a+11b)/16
(11a+21b)/32
(21a+43b)/64
(43a+85b)/128
(85a+171b)/256

An extended sequence of coefficients of a and b and their ratio is:

  coeff. of a     coeff. of b   coeff of a / coeff of b
             1               1  1.00000000000000000000
             1               3  0.33333333333333333333
             3               5  0.60000000000000000000
             5              11  0.45454545454545454545
            11              21  0.52380952380952380952
            21              43  0.48837209302325581395
            43              85  0.50588235294117647059
            85             171  0.49707602339181286550
           171             341  0.50146627565982404692
           341             683  0.49926793557833089312
           683            1365  0.50036630036630036630
          1365            2731  0.49981691688026363969
          2731            5461  0.50009155832265152902
          5461           10923  0.49995422502975373066
         10923           21845  0.50002288853284504463
         21845           43691  0.49998855599551395024
         43691           87381  0.50000572206772639361
         87381          174763  0.49999713898250773905
        174763          349525  0.50000143051283885273
        349525          699051  0.49999928474460375566
        699051         1398101  0.50000035762795391749
       1398101         2796203  0.49999982118608699011
       2796203         5592405  0.50000008940697249216
       5592405        11184811  0.49999995529651775072
      11184811        22369621  0.50000002235174212384
      22369621        44739243  0.49999998882412918788
      44739243        89478485  0.50000000558793546851
      89478485       178956971  0.49999999720603228136
     178956971       357913941  0.50000000139698386322
     357913941       715827883  0.49999999930150806936
     715827883      1431655765  0.50000000034924596556
    1431655765      2863311531  0.49999999982537701728
    2863311531      5726623061  0.50000000008731149138
    5726623061     11453246123  0.49999999995634425432
   11453246123     22906492245  0.50000000002182787284
   22906492245     45812984491  0.49999999998908606358
   45812984491     91625968981  0.50000000000545696821
   91625968981    183251937963  0.49999999999727151589
  183251937963    366503875925  0.50000000000136424205
  366503875925    733007751851  0.49999999999931787897
  733007751851   1466015503701  0.50000000000034106051
 1466015503701   2932031007403  0.49999999999982946974
 2932031007403   5864062014805  0.50000000000008526513
 5864062014805  11728124029611  0.49999999999995736744
 
 

and the denominator is always the sum of the two coefficients.
 
 from

 
  5   point 5
 10   Ap=0:Bp=1
 20   A=1:B=1
 30   for I=1 to 44
 40     An=2*Ap+A:Bn=2*Bp+B
 50     print using(15,0),A,B,using(1,20),A/B
 60     Ap=A:A=An:Bp=B:B=Bn
 70   next

 

Edited on May 23, 2007, 10:11 am

Posted by Charlie on 2007-05-23 10:03:33