An anecdote for GP

Remarks. It may not appear that a geometric growth rate can be very fast, and thus one might be misled to think that the sum of the first \(n\) terms grow slowly. This is far from being the case if the common ratio is more than \(1\). A folklore that illustrates this point runs as follows.

Figure 1. By the time the 5th square is reached, the board contains 31 grains of wheat.

A version of this story was first known to be recorded by Ibn Khallikan in 1256. The farmer who saved his king’s life requested for his ruler to give him wheat according to the rule: “one grain is placed on the first square, two on the second, four on the third and so on, doubling the number of grains of wheat on each subsequent square”. The ruler laughed it off as a meagre prize the farmer was asking for, only to have the court treasurers report the unexpectedly huge number of grains would outstrip the ruler’s resources.

Let \( S_n \) be the total number of grains from the first square to the \( n \)th square. Since this represents the sum of the first \(n \) terms of a geometric progression whose first term is \(1\) and common ratio \( 2 \), we have

\[ S_{64} = 1 + 2 + 2^2 + \cdots + 2^{63} = \frac{2^{64}-1}{2-1} = 18, 446, 744, 073, 709, 551, 615. \]

This indeed is a number of astronomical magnitude.