Введение в последовательности

Imagine a man walking towards a door by first covering half the distance. Then, he covers half of the remaining distance.

Again, half of the new remaining distance is covered, and this process continues.

This pattern creates a sequence of distances. Because there is always a new halfway point, these distances never end, forming an infinite sequence.

This thought experiment is known as Zeno’s paradox, which suggests that if the remaining distance is always halved, the destination is never actually reached.

To analyze such conditions, mathematics introduces the concept of sequence—an ordered list of numbers that follows a specific rule.

Here, each distance is called a term of the sequence, a_n, arranged in a definite order. The distance at any step number n equals one divided by two to the power of n.

As the step number increases, each distance becomes smaller and smaller, but the terms never reach zero.

Because this process continues without end, the sequence is called an infinite sequence.

These sequences help analyze processes involving repetition, change, and infinity.