A GIFT THAT A KING CANNOT DELIVER

There was a chess competition in a king's regime. The king promised gift of winner's choice. After the game, the winner just asked some grains of rice as his price. He added one condition.
He wanted to put one grain of rice on the first square of chess board; 2 grains on the 2nd square; 4 grains on the 3rd; 8 grains on the 4th one. That is, doubling the number of grains each time till reaching 64th square. The king was surprised by the humble request and ordered to execute the prize. The servants brought sack full of rice and begum to arrange the grains on the board as instructed. Halfway through the work, the king and the servants realized that all rice grains in his kingdom would not be enough to fill all the 64 squares. The king submitted to the wisdom of the winner. Then the winner said, "I do not want the price. Just give rice food to the needy daily". And he went away.

This is a good example for the 'geometric series' which is of the form,

= a+ar+ar^2+ar^3+............+ar^n

our example,
=1+1*2+1*4+1*8+............
=1+1*2+1*2^2+1*2^3+.......+1*2^64
The sum of series = a[1-r^n/1-r]
hence, no.of grains= 1*[1-2^64/1-2]
= [1-2^64]/-1 = -1+2^64
= 2^64-1
= 18446744073709551615 grains
=460 billion tones of rice
Moral of the story: the mathematics can kill our ego.

Is 0.999......equal to 1:

0.999....... = 0.9+0.09+0.009+0.0009+..............
= 9/10+9/100+9/1000+.....................
=9/10[1+1/10+1/10^2+1/10^3+.........]-------A
so the series in the bracket is geometric series with a=1,r=1/10 and n goes to infinity
we know,The sum of series = a[1-r^n/1-r]
= 1*[1-(1/10)^infinity/1-1/10]
1/10 to the power infinity logically becomes zero.
than, sum = 1/[1-1/10]
= 1/[9/10] = 10/9
multiply the sum 10/9 with 9/10 which was outside the bracket in equation A
FINALLY,
0.999.......= [9/10]*[10/9] = 1
0.999........definitely is equal to 1

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