Fun with numbers - All about cubes
The cubes have some amazing numerical characteristics.
13 = 1 (one odd )
23 = 3 + 5 = 8 (two successive odds)
= 1x2 +
1 + 2x3 -1
= (22
-1) + (22 = 1)
33 = 7 + 9 + 11 = 27 (three
successive odds)
=( 2x3
+ 1) + ……+(3 x4 -1)
= (32
– 2) + 32 + (32 +2)
43 = 13+15+17+19 = 64 (Four succesivve odds)
=
(3x4+1) +……… +(4x5 -21)
= (42 -3) +(42 – 1) +(42+ 1) + (42 + 3)
53 = 21 + 23 + 25+ 27 + 29 = 125 (Five
successiver odds)
= (4x5
+1) +………….+ )5x6 – 1)
= (52
-4) + (52 – 2) + 52 + (52+ 2) + (52
+ 4)
In general
,n3 = ((n-1)n +1]+ ----------- + [n(n+1)
-1]
=(n2 – n + 1)+ …………..+ (n2 + n -1)
Another property of cubes
13 = 6(0) + 1
23 = 13 + 6(1) + 1 = 6(1) + 2
32 = 23 + 6(3) + 1 = 6(4) + 3
43 = 33 + 6(6) + 1 = 6(10) + 4
53 = 43 + 6(10)+ 1 = 6(20) + 5
63 = 53 + 6(15) + 1 = 6(35) + 6 = 6(36)+ 0
In general
,n3 = (n-1)3 +6 [n(n-1)/2}
+ 1 = 6 [ 1n Σ n(n-1)/2] + n
