Problem
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, what is the total weight in tons of all the wheat that will be placed on the first 59 squares? (Assume that each grain of wheat weighs $1 / 7000$ pound. Remember that 1 ton $=2000 \mathrm{lbs}$.) The total weight of the wheat that will be placed on the first 59 squares is $\square$ tons. (Use scientific notation. Use the multiplication symbol in the math palette as needed. Do not round until the final answer. Then round to six decimal places as needed.) Ask my instructor Clear all Check answer
Solution
Step 1 ：Define the parameters: the first term \(a = 1\), the common ratio \(r = 2\), and the number of terms \(n = 59\).
Step 2 ：Calculate the total number of grains using the formula for the sum of a geometric series: \(total\_grains = a \cdot \frac{r^n - 1}{r - 1}\).
Step 3 ：Convert the total number of grains to pounds: \(total\_pounds = \frac{total\_grains}{7000}\).
Step 4 ：Convert the total number of pounds to tons: \(total\_tons = \frac{total\_pounds}{2000}\).
Step 5 ：The total weight of the wheat that will be placed on the first 59 squares is \(\boxed{41175768021.67311}\) tons.
