A King in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of the 64 squares of a chess board. On the second square the King would place two grains of wheat, on the third square, four grains of wheat, and on the fourth square eight grains of wheat. If the amount of wheat is doubled in this way on each of the remaining squares, how many grains of wheat should be placed on square 19 ? Also find the total number of grains of wheat on the board at this time and their total weight in pounds. (Assume that each grain of wheat weighs $1 / 7000$ pound.)

How many grains of wheat should be placed on square 19 ? grains

How many total grains of wheat should be on the board after the the grains of wheat have been placed on square 19 ?
grains

What is the total weight of all the grains of wheat on the board after the grains of wheat have been placed on square 19 ?
pounds
(Round to the nearest tenth as needed.)

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).