A King in ancient times agreed to reward the inventor of chess with one grain of whea 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 18 ? 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 18 ? 131072 grains
How many total grains of wheat should be on the board after the the grains of wheat have been placed on square 18 ?
grains
Answer
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 \(\boxed{37.449}\).
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 \(\boxed{37.449}\).
