Missing the geometric series

A popular Indian tale involves a craftsman who challenges a king to a game of chess. When he won the game, the king offered him any reward he wished for. The craftsman stated that he wished to have one grain of rice for the first square, two grains for the second, four for the third, eight for the fourth and so on until the 64th square. The king readily grants his wish only to realize later that the sum is a whopping five trillion tonnes of rice – something that we would take about a million years to produce even at current production rates. This story illustrates the insidious power of a geometric series.

Back in middle school, I did not like compound interest. This was because it involved exponents – those unfriendly creatures! Its alternative, simple interest, was true to its name. With simple interest, our investments grow at the same rate each year. With compounding, the meager interest we accrue generates its own interest from the subsequent year. While this might not seem much for a small interest rate, our investments grow surprisingly fast in the long-term. This is because compound interest follows a geometric progression.

With the endeavours we pursue, our returns can be mutually independent, which like simple interest, correspond to an arithmetic series. However, when our efforts add up – when our progress in the second month builds upon that of the first, the results compound. With things we are in for the long-haul – education, careers, fitness and so on, it helps to take a step back every now and then to see if our results are compounding.

As humans, we do not readily think in terms of geometric series. Our intuition, like that of the king in the story, does not recognize their potential until it is too late. And in the process, we do not dig deep enough to see it emerge and reap its rewards.

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