Question

One
gram of wheat on the first of 64 squares of a chess board . On the
second square two grams of wheat , on the third square four grams
of wheat , on the fourth square eight grams of wheat , the amount
doubled for each remaining square, how many grains of wheat should
be placed on square 14 ? Also the total number of grains of wheat
on the board at the time and their total weight in pounds . (Assume
that each grain weights 1/7000 pound)

Answer #1

It is the case of geometric progression. know that if the
1^{st} term of a series is a and if the common ratio
between the successive terms is r, then the nth term of such a
geometric series is ar^{n-1}.

Here, a = 1 and r = 2 so that, the number of grams of wheat on
the 14^{th} square is 1*2^{14-1} = 2^{13} =
8192 grains.

The sum (S_{n}) of the first n terms of a geometric
progression is given by the formula S_{n} =
a(1-r^{n})/(1-r) ( if r ≠ 1), and a and r are as above.

Here, n = 64,a = 1 and r = 2 so that the weight of the wheat on
the chessboard is 1(1-2^{64})/(1-2) = 2^{64} -1
grains = (2^{64} -1)/7000 pounds.

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