Geometry: An Interactive Journey to Mastery

Lesson 34: The Geometry of Figurate Numbers

5. We have proved that
   123   " N NN^2 ^1
   and that
   12         "" NNN 1 1 21 N^2.
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   11 12 13 14
   2122 23 24
   31 32 33 34
   4142 43 44

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7KHVXPRIWKHHQWULHVLQWKH¿UVWURZLVîͼ௘௘ͽDQGWKHVXPRIWKHHQWULHVLQWKHVHFRQGURZLV

îͼ௘௘ͽDQGVRRQ

D௘ͽ ([SODLQZK\WKHVXPRIDOOWKHHQWULHVLQWKHHQWLUHWDEOHHTXDOV

ͼ௘௘ͽ^2.

E௘ͽ 7KHWDEOHFDQEHGLYLGHGLQWRJQRPRQVͼ௘/VKDSHV௘ͽ

11 12 13 14

2122 23 24

31 32 33 34

4142 43 44

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Explain why the sum of the entries in the entire table is also

 îîͼ௘௘ͽîͼ௘௘ͽîͼ௘௘ͽ

F௘ͽ ([SODLQZK\ZHKDYHMXVWSURYHGWKDW^3 + 2^3 + 3^3 + 4^3  ͼ௘௘ͽ^2.

G௘ͽ ([SODLQWKHIRUPXOD 123333  "N^3 NN^24 ^12.