## Tuesday, November 16, 2010

### Odd numbers

1
3       5
7       9      11
13     15     17     19
21   23     25     27     29
31     33     35     37     39     41
43     45     47     49     51     53     55
57    59     61     63     65     67     69     71
73     75     77     79     81     83     85     87     89
91     93     95     97     99    101   103   105   107   109
111   113   115   117   119   121    123    125   127   129   131

Notice the following:

Row #1 : 1 = 1^3
Row #2 : 3 + 5 = 8 = 2^3
Row #3 : 7 + 9 + 11 = 27 = 3^3
Row #4 : 13 + 15 + 17 + 19 = 64 = 4^3
Row #5 : 21 + 23 + 25 + 27 + 29 = 125 = 5^3
Row #6 : 31 + 33 + 35 + 37 + 39 + 41 = 216 = 6^3
Row #7 : 43 + 45 + 47 + 49 + 51 + 53 + 55 = 343 = 7^3
Row #8 : 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 = 512 = 8^3
Row #9 : 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89 = 729 = 9^3
Row #10 : 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109 = 1000 = 10^3
Row #11 : 111 + 113 + 115 + 117 + 119 + 121 + 123 + 125 + 127 + 129 + 131 = 1331 = 11^3

So, in row #n, we would have the sum equal n^3.

Of course, we would need to establish a proof. Would anyone like to prove it?