Geometric sequences are really helpful in our daily lives, especially when we talk about things that grow or shrink.

For example, think about saving money in a bank. If your bank account earns compound interest, the money you have grows each year by a certain percentage. This growth can be shown as a geometric sequence. You start with an initial amount, which is the first number in the sequence. Each year, you get a new amount by multiplying the previous amount by a constant number, called the common ratio.

To find the $n$th term of a geometric sequence, you can use a simple formula:

$$a_n = a_1 \times r^{(n-1)}$$

Here, $a_1$ is your starting amount, $r$ is the common ratio, and $n$ is the position of the term you want to find.

Let’s say you start with £100 and your bank gives you 5% interest. In this case, your common ratio $r$ is 1.05.

Another example of where geometric sequences come in handy is when we look at populations, like bacteria. If one tiny bacterium doubles every hour, you can use geometric sequences to figure out how many bacteria there will be after a certain time.

So, to sum it up: whether you're saving money, estimating how many bacteria will be around, or checking any situation with steady growth or decay, geometric sequences are very useful. They help you make smart decisions based on clear math!