Sequence (A, B, C, D) | (A,B,C) in AP, (B,C,D) in GP

 

Sequence a,b,c,d A1

 
Sequence a,b,c,d A2

 
D - A   ……….   ( \,A, \; B, \; C, \; D \,)

  10   ……….   ( \,6, 9, 12, 16 \,) \; = \; ( \,6, 3^2, 12, 4^2 \,)

  28   ……….   ( \,8, 16, 24, 36 \,) \; = \; ( \,8, 4^2, 24, 6^2 \,)
  28   ……….   ( \,72, 81, 90, 100 \,) \; = \; ( \,72, 9^2, 90, 10^2 \,)

  45   ……….   ( \,4, 16, 28, 49 \,) \; = \; ( \,4, 4^2, 28, 7^2 \,)

  55   ……….   ( \,306, 324, 342, 361 \,) \; = \; ( \,306, 18^2, 342, 19^2 \,)

  91   ……….   ( \,870, 900, 930, 961 \,) \; = \; ( \,870, 30^2, 930, 31^2 \,)

136   ……….   ( \,60, 100, 140, 196 \,) \; = \; ( \,60, 10^2, 140, 14^2 \,)
136   ……….   ( \,440, 484, 528, 576 \,) \; = \; ( \,440, 22^2, 528, 24^2 \,)
136   ……….   ( \,1980, 2025, 2070, 2116 \,) \; = \; ( \,1980, 45^2, 2070, 46^2 \,)

153   ……….   ( \,208, 256, 304, 361 \,) \; = \; ( \,208, 16^2, 304, 19^2 \,)

171   ……….   ( \,270, 324, 378, 441 \,) \; = \; ( \,270, 18^2, 378, 21^2 \,)

190   ……….   ( \,66, 121, 176, 256 \,) \; = \; ( \,66, 11^2, 176, 16^2 \,)
190   ……….   ( \,899, 961, 1023, 1089 \,) \; = \; ( \,899, 31^2, 1023, 33^2 \,)
190   ……….   ( \,3906, 3969, 4032, 4096 \,) \; = \; ( \,3906, 63^2, 4032, 64^2 \,)

253   ……….   ( \,6972, 7056, 7140, 7225 \,) \; = \; ( \,6972, 84^2, 7140, 85^2 \,)

325   ……….   ( \,300, 400, 500, 625 \,) \; = \; ( \,300, 20^2, 500, 25^2 \,)
325   ……….   (11556, 11664, 11772, 11881) \; = \; (11556, 108^2, 11772, 109^2 \,)

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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