Creative thoughts-7

A thought to think
Why people forget the information often which they wanted to convey ?

There are two solid reasons- one is they don't know anything about it. To hide the real state of affair,they are supposed to pretend as if they forgot. Another reason is lack of concentration which causes a lack of confidence.

If two or more pieces of information are stored simultaneously, they are registered randomly and usually not filed in order. Hence they often volatilize during the crucial time of usage

Recreational Mathematics
(Even squares)

4,16,36,64,100,144,196..... are few first even squares. They usually end with 00,4 or 6. .Even squares will always be divisible by 4.

The square of an even number N can be represented by (N/2)[(N-1)+(N+1)], where (N-1) and (N+1) are the two successive odd numbers ,the lower and higher neighbours to the given even number. Thus
2x2 = 1(1+3) = 4 ; 4 x 4 = 2(3+5) = 16; 6 x6 = 3 (5 + 7) = 36 and so on.

It is noted that both the odd and even squares are in the form of 5x or 5x + 1 or 5x - 1.
       odd squares                                             even squares
      2 x 5 - 1 =    9 = 3 x 3                           1 x 5 -  1 =  4  = 2 x 2
    10 x 5 - 1 =  49 = 7 x 7                           3 x 5 + 1 = 16 = 4 x 4
    16 x 5 + 1=  81 = 9 x 9                           7 x 5 + 1 = 36 = 6 x 6
    24 x 5 + 1=121 = 11 x 11                     13 x 5 -  1 = 64 = 8 x 8

Again it is observed that square of a numbe N (may be odd or even) is equal to sum of N successive odd numbers from 1 to (2N-1) in natural series.
                                              1 + 3 = 4 = 2 x 2
                                        1 + 3 + 5 = 9 = 3 x 3
                                1 + 3 + 5 + 7 = 16 = 4 x 4
                          1 + 3 + 5 + 7 + 9 = 25 = 5 x 5
It is seen that the sum of N odd numbers from 1 in the natural series is equal to the square of the mean of all the numbers so added up.
The sum of n even numbers in the natural series of even numbers also has an analogous property. The sum of n successive even numbers from 2 is equal to n times the mean of the even numbers added up.
                       2 = 2 = 1 x 2 = 1 x 1 + 1= 1(1+1)
                 2 + 4 = 6 = 2 x 3 = 2 x 2 + 2  = 2(2+1)
           2 + 4 + 6 = 12 = 3 x4 = 3 x 3 + 3 = 3(3+1)

It is found that the product of any two successive odd or even numbers in the odd or even natural series is equal to one less than the square of the mean of the two numbers

1 x 3 =   3 =   4 - 1 = 2 x 2 - 1   ;    0 x 2 = 0 =    1 - 1 = 1 x 1 - 1
3 x 5 = 15 = 16 - 1 = 4 x 4 - 1   ;    2 x 4 = 8 =    9 - 1 = 3 x 3 - 1
5 x 7 = 35 = 36 - 1 = 6 x 6 - 1   ;    4 x 6 = 24 = 25 - 1 = 5 x 5 - 1