Unveiling Geometric Sequences: Step-by-Step Solutions

by Alex Johnson 54 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of geometric sequences. We'll be tackling a specific problem: figuring out the terms of a geometric sequence given certain conditions. It's like a mathematical treasure hunt, and we're armed with the tools to find the hidden values. Let's break down the problem and uncover the secrets of these sequences. Buckle up, because we're about to embark on a mathematical adventure!

Understanding Geometric Sequences: The Basics

Before we jump into the problem, let's make sure we're all on the same page about geometric sequences. In a nutshell, a geometric sequence is a series of numbers where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio, and it's the key to unlocking the sequence's pattern. Think of it like this: you start with a number (the first term), and then you multiply it by the common ratio to get the next number, and you keep doing this to get the entire sequence.

For example, the sequence 2, 4, 8, 16, and 32 is a geometric sequence. The common ratio is 2 because each term is multiplied by 2 to get the next term. Another example is 100, 50, 25, 12.5, where the common ratio is 0.5. To be very clear, we can see that geometric sequences are fundamental in mathematics and appear in different areas, such as finance (compound interest), computer science (algorithms), and even nature (patterns of growth). The ability to understand and work with geometric sequences is extremely important in the development of math and science.

The beauty of geometric sequences lies in their predictable nature. Once you know the first term and the common ratio, you can find any term in the sequence. This is where the power of formulas comes in. The general formula for the nth term of a geometric sequence is: $a_n = a_1 * r^(n-1)$ where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. We can then use this formula to work our way through all kinds of problems, including the one that we are working on. With this foundation, we can confidently move forward in solving the given conditions of the geometric sequence.

Setting up the Problem: Our Given Conditions

Alright, let's get down to business. In our case, we're given some specific information about a geometric sequence and we need to fill the blanks. Our mission is to find the terms of a geometric sequence that satisfies the given conditions. Let's write down what we know: We are given that $a_4 = 5$, $a_5 = 10$, and $n = 5$. These are our clues, and they will help us unravel the mystery of this geometric sequence. We need to find $a_1$, which is the first term. This is the heart of the problem.

Now, let's think about the relationships between the terms. Remember the definition of a geometric sequence: each term is found by multiplying the previous term by the common ratio. This relationship is very important in this case. Also, it's very important to note that the common ratio (r) is the same between any two consecutive terms. From $a_4$ to $a_5$, we are just multiplying by the common ratio, and therefore we have: $a_5 = a_4 * r$. We can use this to find the common ratio (r).

Finding the Common Ratio (r)

We know that $a_4 = 5$ and $a_5 = 10$. We can use these values to find the common ratio (r). As mentioned earlier, we know the relationship: $a_5 = a_4 * r$. To find r, we simply divide $a_5$ by $a_4$: $r = a_5 / a_4$. Plugging in the values, we get: $r = 10 / 5 = 2$. So, the common ratio (r) is 2. This means that each term in the sequence is multiplied by 2 to get the next term. With the common ratio in hand, we are one step closer to solving the problem. The knowledge of the common ratio is the stepping stone in figuring out other values.

Now, armed with the common ratio, we're in a much better position to find the other terms. The ability to calculate the common ratio is fundamental to solving problems related to geometric sequences. This calculation is a good exercise, and with this approach, you can apply it to other problems.

Determining the First Term ($a_1$)

Now that we've found the common ratio, let's focus on finding the first term, $a_1$. We know $a_4 = 5$ and $r = 2$. Let's go back to our formula: $a_n = a_1 * r^(n-1)$. We can use the information to find $a_1$. Because $a_4$ is the fourth term, then $n = 4$. Replacing the values, we get: $5 = a_1 * 2^(4-1)$. Simplifying, $5 = a_1 * 2^3$. $5 = a_1 * 8$. To isolate $a_1$, we simply divide 5 by 8: $a_1 = 5 / 8$. Therefore, the first term $a_1$ is 5/8 or 0.625.

Now we have all the ingredients to write the sequence. We've found the common ratio and the first term. This allows us to find any term of the sequence. It's like putting the pieces of a puzzle together. With the first term and the common ratio in hand, we have everything we need to define the geometric sequence. Now that we have found the first term, we can find the rest of the terms.

Writing out the Terms of the Geometric Sequence

We've done it! We have found the first term ($a_1 = 5/8$) and the common ratio (r = 2). Now, let's write out the terms of the geometric sequence. We know we want to find the first five terms (n = 5). Let's calculate each term using the formula $a_n = a_1 * r^(n-1)$:

  • First term ($a_1$): We already know this is 5/8 or 0.625.
  • Second term ($a_2$): $a_2 = (5/8) * 2^(2-1) = (5/8) * 2 = 10/8 = 1.25$
  • Third term ($a_3$): $a_3 = (5/8) * 2^(3-1) = (5/8) * 4 = 20/8 = 2.5$
  • Fourth term ($a_4$): We're given that this is 5.
  • Fifth term ($a_5$): We're given that this is 10.

So, the geometric sequence is: 5/8, 10/8, 20/8, 5, 10 (or 0.625, 1.25, 2.5, 5, 10). We successfully found the terms of the geometric sequence! We have solved the problem, and we know that we can do this for any other similar problem. We've shown how to use the given information to find the common ratio and the first term. These values unlock the entire sequence. We also demonstrated the power of the formula to find other terms.

Conclusion: Mastering Geometric Sequences

Congratulations, you've successfully navigated the world of geometric sequences! We've taken a problem with given conditions and, step by step, figured out how to find the first term, the common ratio, and the subsequent terms of the sequence. Remember, the key is to understand the concept of a constant ratio and the use of the formula: $a_n = a_1 * r^(n-1)$. By practicing these problems, you'll become more comfortable in working with geometric sequences.

Geometric sequences are used in different areas of mathematics, and understanding them is a step forward in our journey of math. Keep practicing and keep exploring the amazing world of math. You've got this!

For further exploration, you might find this website helpful: Khan Academy - Geometric Sequences. This link will redirect you to a trusted website related to the topic.