[math-fun] Proofs without words
Can someone please point me to proofs without words for a) nth tetrahedral number = n(n+1)(n+2)/6 b) sum of the first n squares = n(n+1)(2n+1)/6 c) sum of the first n cubes = (sum of first n numbers)^2 = (T_n)^2 (T_n = nth triangular #) ? Thanks, Dan
a) Subtract (n-1) x n x (n+1) = n x (n+1) x (n-1) cuboid from one end of n x (n+1) x (n+2) cuboid, leaving n x (n+1) rectangle x 3 = n-triangle x 6 --- a visually appealing diagram would require some draughtsmanship I imagine, certainly more than is readily available in this text-file! c) More of a challenge --- looks as if 4 space dimensions would be required ... WFL On 12/31/07, Dan Asimov <dasimov@earthlink.net> wrote:
Can someone please point me to proofs without words for
a) nth tetrahedral number = n(n+1)(n+2)/6
b) sum of the first n squares = n(n+1)(2n+1)/6
c) sum of the first n cubes = (sum of first n numbers)^2 = (T_n)^2 (T_n = nth triangular #) ?
Thanks,
Dan
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Can someone please point me to proofs without words for c) sum of the first n cubes = (sum of first n numbers)^2
I love this one. I'll try to ascii-fy it: A BB CCC DDDD B EE FFF GGGG B EE FFF GGGG C FF HHH IIII C FF HHH IIII C FF HHH IIII D GG III JJJJ D GG III JJJJ D GG III JJJJ D GG III JJJJ A BB EE BB EE CCC FFF HHH FFF FFF HHH FFF CCC HHH DDDD GGGG IIII JJJJ IIII GGGG IIII JJJJ IIII GGGG IIII JJJJ IIII GGGG DDDD JJJJ I think JHC & RKG's _Book of Numbers_ has a nice 3-d picture of this, but my copy isn't to hand to check. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
See Conway & Guy, The Book of Numbers. On Mon, 31 Dec 2007, Dan Asimov wrote:
Can someone please point me to proofs without words for
a) nth tetrahedral number = n(n+1)(n+2)/6
b) sum of the first n squares = n(n+1)(2n+1)/6
c) sum of the first n cubes = (sum of first n numbers)^2 = (T_n)^2 (T_n = nth triangular #) ?
Thanks,
Dan
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participants (4)
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Dan Asimov -
Fred lunnon -
Michael Kleber -
Richard Guy