Some sequences build by adding the same amount each time. Others grow by multiplying. These are called geometric sequences. Instead of a common difference, they have a common ratio.

Consider the sequence 3, 6, 12, 24, ... Each number is twice the one before it. The number we keep multiplying by, in this case 2, is the common ratio. We usually label it with the letter $r$.

The common ratio doesn't have to be a whole number. It can be a fraction, a negative number, or even a decimal. Look at this sequence:

100, 50, 25, 12.5, ...

Here, the common ratio is $50 \div 100 = 0.5$. Each term is half of the previous one.

And this one:

5, -15, 45, -135, ...

The common ratio is $-15 \div 5 = -3$. The terms alternate between positive and negative.

What if you need to find the 20th term of a sequence? You wouldn't want to multiply it out 19 times. Luckily, there's a formula for that. Let's figure it out.

Let's call the first term $a_1$ and the common ratio $r$. The sequence looks like this:

• The 1st term is $a_1$.
• The 2nd term is $a_1 \cdot r$.
• The 3rd term is $(a_1 \cdot r) \cdot r = a_1 \cdot r^2$.
• The 4th term is $(a_1 \cdot r^2) \cdot r = a_1 \cdot r^3$.

Notice a pattern? The power of $r$ is always one less than the term number. This gives us a general formula to find any term, which we call the $n$-th term, or $a_n$.

Let's use this formula. We want to find the 8th term of the sequence 4, 12, 36, ...

First, identify what we know:

• The first term, $a_1$, is 4.
• The common ratio, $r$, is $12 \div 4 = 3$.
• The term number we want, $n$, is 8.

Now, we plug these values into the formula.

So, the 8th term of the sequence is 8,748. That was much faster than multiplying by 3 seven times.

Geometric sequences change quickly. If the common ratio is greater than 1, the terms grow exponentially. If the ratio is between 0 and 1, they shrink exponentially towards zero. The graph of a geometric sequence isn't a straight line; it's a curve.

This visual shows how quickly the terms can shoot up or drop off. Understanding this behavior is key to seeing how geometric sequences apply to real-world situations like population growth or radioactive decay.

Geometric sequences are a fundamental pattern in mathematics, defined by their constant multiplicative growth or decay.