Chapter 4, Problem 40C

Explanation of Solution

Total number of grains requested by the inventor of chess:

If a chessboard is placed with square then one grain is placed on first square, two on second, and four on third. On each subsequent square, the numbers of grains are doubled.

Thus, the exponential increase on each square contains $2^{n-1}$ grains of rice.

Since, there are 64 squares on the chessboard, apply the value of 64 in the formula $2^{n-1}$ to fill the final square on the chessboard. Substitute “n” as 64.

  $\begin{array}{c}Fillingofriceonfinalsquareonchessboard=2^{64-1}\\ =2^{63}\\ =9223372036854775808\\ =9.22\times {10}^{18}\end{array}$

To determine the total number of grains of rice let us use the exponential sum using the formula $2^n-1$