Creative thoughts-8

Creative thoughts-8


If our goal and desire is pale and anemic, it will invariably be reflected in all our
activities. Yes, all the deeds done outside are simply the follow-up of the initial
and inherent ambitions inside. If the later is good, the former will also be good.
If we go after our subject with persistence we can reach the target like a
forward wave from a light source that reaches the obstacle at infinite distance
with non-stop propagation

Recreational Mathematics.

Fourth power of a number

* The fourth power of a number N can be expressed as the sum of the product of
(NxN -1) and (NxN + 1) and 1 i.e.,

NxN xN xN = (NxN-1)(NxN+1) + 1

* The fourth power of even numbers is equal to a multiple of 16 and one excess
over a multiple of 16 for all odd numbers.

1x1x1x1 =      1 =  0 x 16 +  1  ; 2x2x2x2 =     16 = 16 x 1 + 0
3x3x3x3 =    81 =  5 x 16 +  1  ; 4x4x4x4 =   256 = 16 x16 +0
5x5x5x5 =  625 = 39 x16 +  1  ; 6x6x6x6 = 1296 = 16 x 81 +0
7x7x7x7 = 2401 =150 x16 +1  ; 8x8x8x8 = 4096 = 16 x 256 + 0

It shows that NxNxNxN – 1 is completely divisible by 16 ,when N is add..Again,

NxNxNxN – 1 = (NxN+1)(N-1)(N+1)

* When N is odd, all these factors will be even and hence NxNxNxN – 1 will be
divisible by 8.

* The fourth power of a number which is not a multiple of 5, is equal to one excess
over a multiple of 5.

2x2x2x2 =   16 = 1 x 5 + 1  ; 6x6x6x6 =   1296 =   259 x5 +1
3x3x3x3 =   81 = 16x5 + 1  ; 7x7x7x7 =   2401 =   480 x5 + 1
4x4x4x4 = 256 = 51x5 + 1  ;  8x8x8x8 =  4096 =   819 x5 + 1
                                            ; 9x9x9x9 =   6561 = 1312 x 5 + 1

* The fourth power of a number N can be expressed as the sum of NxN odd numbers from 1.

1x1x1x1 = 1
2x2x2x2 = 1+3+5+7 = 16
3x3x3x3 = 1+3+5+7+9+11+13+15+17 = 81
4x4x4x4 = 1+3 ……….29 +31 = 256







NxN

NxNxNxN = ∑ (2n-1)

1