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Sequence of wrong means (Posted on 2011-05-26) Difficulty: 2 of 5
A) Start a sequence with any two real numbers, for the third and successive terms take the mean of all of the previous terms.

B) Same as part A) but take the mean incorrectly as follows: add the numbers together but instead of dividing by the number of terms, divide by one less than the number of terms.

C) Same as part A) and B) but take the mean incorrectly as follows: add the numbers together but instead of dividing by the number of terms, divide by one more than the number of terms.

What happens to the sequence in each case? Generalize further.

See The Solution Submitted by Jer    
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Solution apparently the solution | Comment 1 of 3

As expected, when the true means are used, the first mean found is the mean of the two initial numbers, and thereafter the mean stays the same:

DO
  INPUT a, b
  tot = a + b
  FOR n = 2 TO 45
    newt = tot / (n)
    PRINT newt
    tot = tot + newt
  NEXT
LOOP


? 2,5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5
 3.5

What is less readily understood is that the means are also stable in the other two conditions:

DO
  INPUT a, b
  tot = a + b
  FOR n = 2 TO 45
    newt = tot / (n-1)
    PRINT newt
    tot = tot + newt
  NEXT
LOOP

? 2,5
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 
 so in part B, the "mean" is the sum of the two numbers, and remains such after each generation of "mean".
 
 In part C the first "mean" and all subsequent ones is 1/3 the sum of the first two:
 
DO
  INPUT a, b
  tot = a + b
  FOR n = 2 TO 45
    newt = tot / (n + 1)
    PRINT newt
    tot = tot + newt
  NEXT
LOOP

? 2,5
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333333
 2.333334
 2.333334
 2.333334
 2.333334
 2.333334
 2.333334
 2.333334
 2.333334
 2.333334


  Posted by Charlie on 2011-05-27 02:04:40
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