Step 1 :The problem is asking for the number of grains on the 19th square, the total number of grains on the board after the 19th square, and the total weight of the grains.
Step 2 :The number of grains on each square is a geometric progression where the first term is 1 (one grain on the first square) and the common ratio is 2 (the number of grains doubles on each square).
Step 3 :The number of grains on the nth square can be calculated as \(a_n = a_1 * r^{(n-1)}\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
Step 4 :The sum of the first n terms of a geometric progression can be calculated as \(S_n = a_1 * (1 - r^n) / (1 - r)\), where \(S_n\) is the sum of the first n terms, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
Step 5 :The weight of the grains can be calculated by multiplying the total number of grains by the weight of one grain, which is given as \(1 / 7000\) pound.
Step 6 :Using these formulas, we find that the number of grains on the 19th square is \(a_1 * r^{(19-1)} = 262144\) grains.
Step 7 :The total number of grains on the board after the grains of wheat have been placed on square 19 is \(a_1 * (1 - r^{19}) / (1 - r) = 524287\) grains.
Step 8 :The total weight of all the grains of wheat on the board after the grains of wheat have been placed on square 19 is \(524287 * (1 / 7000) = 74.9\) pounds (rounded to the nearest tenth).
Step 9 :Final Answer: The number of grains of wheat that should be placed on square 19 is \(\boxed{262144}\) grains. The total number of grains of wheat on the board after the grains of wheat have been placed on square 19 is \(\boxed{524287}\) grains. The total weight of all the grains of wheat on the board after the grains of wheat have been placed on square 19 is \(\boxed{74.9}\) pounds (rounded to the nearest tenth).