How to Solve a Geometric Sequence: Step-by-Step Guide
Quick Answer
To solve a geometric sequence, identify the first term and the common ratio. Use the formula a_n = a_1 * r^(n-1) to find any term in the sequence.
Understanding geometric sequences is essential in math, and it can be easier than you think. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Let's break down how to solve a geometric sequence step-by-step.
**Step 1: Identify the First Term**
The first term of a geometric sequence is usually denoted as a_1. For example, if the sequence starts with -1, then a_1 = -1. This value will be the starting point for your calculations.
**Step 2: Find the Common Ratio**
The common ratio (r) is what you multiply each term by to get to the next term. To find r, take the second term and divide it by the first term. Continuing with our example, if the second term is 5, then:
-5 = 5 / (-1)
From this, we can see that r = -5. Check this with the next term: if the third term is -25, then confirm that 5 * (-5) = -25. So, the common ratio is indeed -5.
**Step 3: Write the General Formula**
The formula for a geometric sequence is:
$$a_n = a_1 * r^{(n-1)}$$
Here, a_1 is the first term and r is the common ratio. Substituting in our values gives us:
a_n = -1 * (-5)^{(n-1)}.
This means that to find any term in the sequence, you can plug in n (where n is the term number).
**Step 4: Verify the Formula**
It's always a good idea to check your work. Let's see if our formula holds true for the first few terms:
- For n = 1:
a_1 = -1 * (-5)^(0) = -1 * 1 = -1
- For n = 2:
a_2 = -1 * (-5)^(1) = -1 * (-5) = 5
- For n = 3:
a_3 = -1 * (-5)^(2) = -1 * 25 = -25
This confirms that our formula is correct!
**Real-World Applications**
Geometric sequences can be found in various real-world situations such as population growth, financial calculations involving compound interest, and even in certain patterns in nature. Understanding how to work with these sequences can provide valuable insights and skills applicable in many fields.
By following these steps, you can confidently solve any geometric sequence. Whether you're studying for a test or just want to improve your math skills, mastering this concept is a great way to train your brain!
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