Step 1 :The problem is asking for the number of grains of wheat on the 18th square of a chessboard, where the number of grains doubles on each square. This is a geometric progression problem, where the first term is 1 (one grain on the first square), the common ratio is 2 (the number of grains doubles each time), and we want to find the 18th term.
Step 2 :The formula for the nth term of a geometric progression is \(a*r^{(n-1)}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
Step 3 :So, to find the number of grains on the 18th square, we can plug in the values into the formula: \(1*2^{(18-1)} = 2^{17}\).
Step 4 :To find the total number of grains on the board after the grains have been placed on the 18th square, we can use the formula for the sum of the first n terms of a geometric progression: \(a*(r^n - 1)/(r - 1)\). Plugging in the values, we get: \(1*(2^{18} - 1)/(2 - 1) = 2^{18} - 1\).
Step 5 :Finally, to find the total weight of the grains in pounds, we can multiply the total number of grains by the weight of each grain (1/7000 pounds).
Step 6 :Let's calculate these values: the number of grains on the 18th square is \(2^{17} = 131072\), the total number of grains on the board is \(2^{18} - 1 = 262143\), and the total weight of the grains in pounds is approximately \(262143 / 7000 = 37.449\).
Step 7 :Final Answer: The number of grains of wheat that should be placed on square 18 is \(\boxed{131072}\). The total number of grains of wheat on the board after the grains of wheat have been placed on square 18 is \(\boxed{262143}\). The total weight of the grains in pounds is approximately \(\boxed{37.449}\).