Grains on the chessboard and other high powers

Here is a classic exercise that can be tackled at different stages of education: how many grains of wheat do we get by placing one grain on the 1^st square of a chessboard, two on the 2^nd, four on the 3^rd, eight on the 4^th, and so on, doubling the number of grains up to the 64^th and final square?

This puzzle, which may have originated in Persia, has a history stretching back more than a thousand years and comes in countless versions. From a purely mathematical standpoint, its most fascinating feature is the enormity of the answer: 18,446,744,073,709,551,615. Try saying it aloud on your first attempt without hesitating!

The number of digits in the final answer means that mental arithmetic alone cannot produce it, unless you are a calculating prodigy. Yet it is possible to work out an extremely good approximation in your head—one that shows that, no, large numbers are not beyond our reach.

Tackling the sum ---------------------