Chapter 12.3, Problem 95AYU

To determine

To calculate: The total number of wheat grains needed to fill the chess board as per the commoner wish.

Answer to Problem 95AYU

The total number of wheat grains $S_{64}=1.845\times {10}^{19}$.

Explanation of Solution

Given:

Commoner wish to fill the chessboard using wheat grains. He wants to place one grain of wheat on the first square of the chessboard, two grains on second square, four grains on the third, and so on.

Calculation:

The sequence is $1,2,4,8,16,32,64,\cdots$.

The first term is $a_1=1$. The common ratio $r=\frac{a^n}{a^{n-1}}=2$.

Therefore, the given sequence is geometric since the ratio of successive term is 2.

There are 64 squares in the chessboard. So the total number of grains needed to fill this 64 square becomes $S_{64}$.

Sum of $n$ terms of an geometric sequence $=S_n=a_{1 }\frac{1-r^n}{1-r}$.

Sum of 64 terms $=S_{64}=1\times \frac{1-{\left(2\right)}^{64}}{1-2}=-1\left(1-{\left(2\right)}^{64}\right)={\left(2\right)}^{64}-1=1.845\times {10}^{19}$.