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Add 1, Get Harmonic (Posted on 2013-01-25) Difficulty: 3 of 5
Determine all possible triplets (x,y,z) of positive integers with x < y < z < 100 such that: x, y and z (in this order) are in arithmetic sequence and x, y and z+1 are in harmonic sequence.

Bonus Question:

Amending the restriction to x < y < z, prove that there are an infinity of positive integer triplets satisfying the given conditions.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Some Thoughts computer exploration Comment 2 of 2 |

DECLARE FUNCTION gcd! (x!, y!)
DECLARE FUNCTION lcm! (x!, y!)
FOR x = 1 TO 998
 FOR incr = 1 TO 600
  IF x + 2 * incr < 999 THEN
   y = x + incr
   z = y + incr
   comDen = lcm(lcm(x, y), z + 1)
   a = comDen / x
   b = comDen / y
   c = comDen / (z + 1)
   IF b - a = c - b THEN
     PRINT x - px; incr - pincr, x; y; z; TAB(40); a; b; c
     px = x: pincr = incr
   END IF
  END IF
 NEXT
NEXT

FUNCTION gcd (x, y)
 dnd = x: dvr = y
 IF dnd < dvr THEN SWAP dnd, dvr
 DO
   q = INT(dnd / dvr)
   r = dnd - q * dvr
   dnd = dvr: dvr = r
 LOOP UNTIL r = 0
 gcd = dnd
END FUNCTION

FUNCTION lcm (x, y)
 lcm = x * y / gcd(x, y)
END FUNCTION


finds

  delta                       verification of
  x incr     x    y    z   inverses in arithmetic seq.
  3  1       3    4    5         4   3   2
  7  1      10   12   14         6   5   4
 11  1      21   24   27         8   7   6
 15  1      36   40   44        10   9   8
 19  1      55   60   65        12  11  10
 23  1      78   84   90        14  13  12
 27  1     105  112  119        16  15  14
 31  1     136  144  152        18  17  16
 35  1     171  180  189        20  19  18
 39  1     210  220  230        22  21  20
 43  1     253  264  275        24  23  22
 47  1     300  312  324        26  25  24
 51  1     351  364  377        28  27  26
 55  1     406  420  434        30  29  28
 59  1     465  480  495        32  31  30
 63  1     528  544  560        34  33  32
 67  1     595  612  629        36  35  34
 71  1     666  684  702        38  37  36
 75  1     741  760  779        40  39  38
 79  1     820  840  860        42  41  40
 83  1     903  924  945        44  43  42

The first six are the ones under 100 that are asked for.

The arithmetic progression of the delta-x's indicates that the x values have a quadratic relationship to the line number, and some algebra shows that x = 2n^2 + n is the relation between the line number n, and x. The increment increases by 1 for each line, and is just n.

So x, y and z are:

n*(2*n+1), n*(2*n+2) and n*(2*n+3)

The reciprocals that go into confirming the harmonic progression are:

1/(n*(2*n+1)), 1/(n*(2*n+2)) and 1/(n*(2*n+3)+1)

which of course have to be converted to a common denominator, simplified and scaled to give the numbers shown above on the right.


  Posted by Charlie on 2013-01-25 12:34:46
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