When you enter the fun world of geometric sequences, you’ll come across some important formulas. These will help you find any term in the sequence and add them up easily! Let’s break down these tools so you can understand geometric sequences better.

What is a Geometric Sequence?

First, let’s talk about what a geometric sequence is. A sequence is geometric if you can get each term after the first by multiplying the previous term by a certain number. This number is called the common ratio ($r$).

For example, if we start with 2 and our common ratio is 3, the geometric sequence would look like this:

• First term ($a_1$): 2
• Second term ($a_2$): $2 \cdot 3 = 6$
• Third term ($a_3$): $6 \cdot 3 = 18$
• Fourth term ($a_4$): $18 \cdot 3 = 54$

So, the sequence is 2, 6, 18, 54, and it keeps going!

Finding the Nth Term

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

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

Where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the term number.

Example:

Let’s say our first term $a_1$ is 5 and our common ratio $r$ is 4. To find the 6th term ($n = 6$), we do this:

$$a_6 = 5 \cdot 4^{(6-1)} = 5 \cdot 4^5$$

Calculating $4^5$ gives us 1024, so:

$$a_6 = 5 \cdot 1024 = 5120$$

This means the 6th term is 5120.

Finding the Sum of the First N Terms

If you want to find the sum of the first $n$ terms of a geometric sequence, use this sum formula:

$$S_n = \frac{a_1(1 - r^n)}{1 - r} \quad \text{(for } r \neq 1\text{)}$$

Here, $S_n$ is the sum of the first $n$ terms.

Example:

Let’s find the sum of the first 4 terms from our earlier example where $a_1 = 5$ and $r = 4$:

$$S_4 = \frac{5(1 - 4^4)}{1 - 4}$$

Calculating $4^4$ gives us 256, so:

$$S_4 = \frac{5(1 - 256)}{-3} = \frac{5(-255)}{-3} = \frac{1275}{3} = 425$$

So, the sum of the first 4 terms is 425.

Conclusion

In short, remember these key formulas to work with geometric sequences. With a little practice, you’ll be solving these problems easily and impressing your friends in no time!