Geometric Sequence: Find The Second Term (a2)

Nov 19, 2025 by Andrew McMorgan 46 views

Finding the Second Term of a Geometric Sequence: A Step-by-Step Guide

Hey guys! Today, let's dive into the fascinating world of geometric sequences and tackle a common problem: finding a specific term when you're given a recursive formula. Recursive formulas can seem a little intimidating at first, but trust me, they're super manageable once you understand the basic idea. We'll break down a problem step-by-step so you can conquer these types of questions with confidence. So, grab your thinking caps, and let's get started!

Understanding Geometric Sequences and Recursive Formulas

Before we jump into solving the problem, let's quickly recap what geometric sequences and recursive formulas are all about. This foundational knowledge is crucial for understanding the problem and the solution. A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value, called the common ratio. Think of it like this: you start with a number, and then you multiply it by the same value over and over again to get the next numbers in the sequence. That constant multiplier is our common ratio.

Now, what's a recursive formula? Well, it's a way of defining a sequence where each term is defined in relation to the terms that come before it. In other words, to find a term, you need to know the value of the previous term (or terms). This is different from an explicit formula, where you can directly calculate any term by plugging in its position in the sequence. Recursive formulas are like following a set of instructions step-by-step, using the previous result to get the next one. This makes them really useful for describing sequences where the pattern depends on the terms that have already been generated. So, remember, geometric sequences involve a common ratio, and recursive formulas define terms based on previous terms. With these concepts in mind, we're ready to tackle our problem!

The Problem: Decoding the Recursive Formula

Okay, let's get down to business! We're given a recursive formula for a geometric sequence: $a_1 = 5$ and $a_n = rac{2}{5}a_{n-1}$ . Our mission, should we choose to accept it (and we do!), is to find the second term of this sequence, which we denote as $a_2$ . The first part of the formula, $a_1 = 5$ , is our starting point. It tells us that the first term of the sequence is 5. This is our initial value, the foundation upon which we'll build the rest of the sequence. The second part, $a_n = rac{2}{5}a_{n-1}$ , is the heart of the recursive definition. It tells us how to find any term ( $a_n$ ) based on the term that comes before it ( $a_{n-1}$ ). The $rac{2}{5}$ is the common ratio, the magic number that links each term to the next. It's the key to unlocking the sequence's pattern.

To find the second term ( $a_2$ ), we need to use this recursive part of the formula. We'll substitute $n = 2$ into the formula, which will give us an equation that relates $a_2$ to $a_1$ . Since we already know $a_1$ , we can then easily solve for $a_2$ . So, let's take a closer look at how we'll apply this recursive formula to find the term we're after. Remember, the beauty of recursive formulas lies in their step-by-step nature, allowing us to build the sequence one term at a time. Keep this in mind, and you'll find these problems much less daunting! Now, let's put this understanding into action and calculate $a_2$ .

Step-by-Step Solution: Calculating the Second Term ( $a_2$ )

Alright, let's roll up our sleeves and get to the actual calculation! We want to find the second term, $a_2$ , and we have the recursive formula $a_n = rac{2}{5}a_{n-1}$ . The first step is to substitute $n = 2$ into the formula. This is where the magic happens, where we connect the general formula to the specific term we're interested in. When we replace $n$ with 2, we get: $a_2 = rac{2}{5}a_{2-1}$ . Notice how we've simply swapped the $n$ with 2 in the formula. Now, let's simplify the expression inside the subscript: $2 - 1 = 1$ . So, our equation becomes: $a_2 = rac{2}{5}a_1$ .

See how this connects the second term, $a_2$ , to the first term, $a_1$ ? This is exactly what the recursive formula is designed to do. Now, we're in the home stretch! We know from the given information that $a_1 = 5$ . So, we can substitute this value into our equation: $a_2 = rac{2}{5} imes 5$ . Now it's just a matter of simple multiplication. We can think of this as $rac{2}{5}$ multiplied by $rac{5}{1}$ , which gives us $rac{2 imes 5}{5 imes 1} = rac{10}{5}$ . Finally, we simplify the fraction: $rac{10}{5} = 2$ . Therefore, the second term of the sequence, $a_2$ , is 2. And there you have it! We've successfully used the recursive formula to find the second term. Wasn't that satisfying? Now, let's move on to solidifying our understanding by looking at the answer choices.

Identifying the Correct Answer

Now that we've done the hard work and calculated that $a_2 = 2$ , it's time to check our answer against the given options. This is a crucial step, guys! It's always a good idea to double-check your work and make sure you're selecting the right answer. It’s super easy to make a small mistake in the calculation, so verifying is a smart move. Looking at the options provided, we have:

A. $a_2 = 2$
B. $a_2 = 5 rac{2}{5}$
C. $a_2 = rac{8}{5}$
D. $a_2 = rac{2}{5}$

It's pretty clear that option A, $a_2 = 2$ , matches our calculated answer perfectly! The other options are all different values, so we can confidently eliminate them. Option B is a mixed number, which is clearly not equal to 2. Options C and D are fractions, and while it's always a good idea to quickly check if they could simplify to 2 (which they don't), we can see that they're not the correct answer. So, the moment of truth! We've confirmed that our calculated answer matches one of the options. This gives us a great sense of accomplishment and reinforces our understanding of the problem-solving process. Remember, correctly identifying the answer is just as important as doing the calculation itself. Now, let's give ourselves a pat on the back and move on to the final section where we'll recap the key takeaways and reinforce what we've learned.

Key Takeaways and Practice Problems

Awesome job, guys! You've successfully navigated the world of recursive formulas and geometric sequences. Let's take a moment to recap the key takeaways from this problem. First and foremost, we learned how to use a recursive formula to find a specific term in a sequence. The trick is to understand that the formula defines each term based on the preceding term(s). We saw how substituting the appropriate value for n allows us to connect the term we want to find to the term we already know. Secondly, we reinforced the importance of careful calculation and double-checking our answer against the given options. It's so easy to make a small arithmetic error, so taking that extra moment to verify can make all the difference. Finally, we solidified our understanding of geometric sequences and how they are defined by a common ratio. Remember, this constant multiplier is the backbone of the sequence's pattern.

To further solidify your understanding, here are a couple of practice problems you can try on your own:

Given the recursive formula $a_1 = 3$ , $a_n = 2a_{n-1}$ , find $a_3$ .
A geometric sequence is defined by $a_1 = -2$ , $a_n = - rac{1}{2}a_{n-1}$ . Find $a_2$ and $a_4$ .

Tackling these problems will give you more hands-on experience and build your confidence in working with recursive formulas and geometric sequences. Remember, practice makes perfect! Keep exploring these concepts, and you'll become a pro in no time. You can do it!