'Geometric' is perhaps the wrong term for a sequences with terms defined by the ruleEach term is being multiplied by a constant factor to obtain the next term.

A quantity is said to grow or decay exponentially if the quantity at the start of each time period is multiplied by a constant factor to obtain the quantity at the end of the period, or the start of the next period.

If the constant factor is less than 1, then the quantity is decaying exponentially.

If the quantity is greater than 1, the quantity is growing exponentially.

Exponential growth is what the World economy has been experiencing for the past several hundred years. Every year since 1700 the world economy has grown by about 2.1% on average. 2.1% does not sound like a large increase, but a growth rate of 2.1% means that the World economy has been doubling every 33 years.

Economies have always grown by finding new lands to conquer and resources to exploit. Now however, all the land on Earth essentially belongs to someone, so no - one can gain title by just sitting on it and shouting, 'this is mine'. Essentially all the natural resources on Earth are being exploited or overexploited. The future we face can be illustrated by considering oil. Roughly half of all the oil that was in the ground has been exploited. Demand for oil is growing year on year, but there are estimated to be only about 40 years worth of recoverable oil reserves in the ground. With no feasible alternative to oil in sight, the World economy faces a crunch when available supplies of oil inevitably dry up.

Suppose then, that astronomers found a 'twin Earth' on the other side of the Sun, and scientists could work out how to exploit it for its oil. Would we be saved? No. Assuming oil consumption grows at 2.1% a year, and in the current year 1/40 of the Earth's oil reserves are used up, equal to 1/120 of our augmented reserves, the 1.5 'Earth's worth of oil' now available to us would only last n years, where n is the solution to the equation

Rearrangement givesyears.