Wheat on the chessboard

The legend puts $1$ grain on the first square, $2$ on the second, $4$ on the third, doubling each time. Square $n$ holds $2^{n-1}$ grains. Summing the whole board is a finite geometric series: $1 + 2 + 4 + \dots + 2^{63}$.

This is geometric growth: each term is a fixed multiple of the last, and the sum of $1 + r + r^2 + \dots + r^{m}$ is $\frac{r^{m+1} - 1}{r - 1}$. For $r = 2$ that formula collapses to a single power of two minus one. Doubling explodes because the growth compounds on an ever-larger base.