Geometric Sequence

Geometric sequences are another fundamental and important type of sequence. Unlike arithmetic sequences with their “linear growth,” geometric sequences exhibit “exponential growth”—this growth pattern is ubiquitous in nature and human society.

From bacterial reproduction to compound interest calculations, from radioactive decay to viral spread, geometric sequence models help us understand these seemingly different phenomena that are essentially the same.

The Power of Geometric Sequences

There’s a famous story about the inventor of chess who asked the king for a reward. He requested 1 grain of rice on the first square, 2 grains on the second, 4 on the third, and so on, with each square having double the rice of the previous one. The king thought this was simple and agreed.

But when calculated, the total rice needed for all 64 squares was:

$1 + 2 + 4 + 8 + \cdots + 2^{63} = 2^{64} - 1 \approx 1.84 \times 10^{19}$

This number exceeds the total amount of all grain in the world! This is the amazing power of geometric sequences (exponential growth).