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 = 2¹³ = 8192 grains.

The sum (S_n) of the first n terms of a geometric progression is given by the formula S_n = a(1-rⁿ)/(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(1-2⁶⁴)/(1-2) = 2⁶⁴ -1 grains = (2⁶⁴ -1)/7000 pounds.