Consider this infinite sequence:

          a, b, c, (a+b+c)/3, (a+4b+4c)/9, (4a+7b+16c)/27 .....

where, each of a, b and c is a real number and, each term after the third is the arithmetic mean of the three previous terms.

Determine the limit of the sequence in terms a, b and c.

DEFDBL A-Z
CLS
DATA 2,1,1
DATA 1,2,1
DATA 1,1,2
DATA 1,2,5
DATA 2,3,13
DO
  READ a, b, c
  FOR i = 1 TO 45
    d = (a + b + c) / 3
    a = b: b = c: c = d
  NEXT
  PRINT d
LOOP

finds
                             for
                       a      b      c
1.166666666664709      2      1      1
1.333333333340315      1      2      1
1.499999999994976      1      1      2
3.333333333320219      1      2      5
7.833333333285052      2      3     13

after 45 iterations beyond c.

Assuming the limit values are 7/6, 4/3, 3/2 for the first three and that there's a linear relationship between a, b, c and the limit, it would seem that the limit is (1/2 * a + b + 3/2 * c) / 3, and this is confirmed by the remaining two sets of a,b,c.

Posted by Charlie on 2013-09-24 13:26:21