Chapter 12.3, Problem 87AYU

To determine

To find: The total number of grains.

Answer to Problem 87AYU

The total number of grains is $1.845\times {10}^{19}$ .

Explanation of Solution

Given information:

The given dimension of chess board is $8\times 8$ .

Calculation:

The total number of grains is $1+2+4+\mathrm{8...........}$

This is a geometric sequence with the ratio of the consecutive terms as $2$ .

Chess consists of $64$ . So, the term in the geometric sequence is $64$ .

The sum of $n$ term of the geometric sequence is $S_n=a_1.\frac{1-r^2}{1-r}$ , here $n=64,r=2\text{ and }{a}_1=1$ .

Substitute the value.

  $\begin{array}{c}{S}_n=1.\frac{1-2^{64}}{1-2}\\ =1.845\times {10}^{19}\end{array}$

Therefore, the total number of grains is $1.845\times {10}^{19}$ .