Is $a_n = 5n^2 + 4$ Geometric? Let's Find Out!

Hey guys! Today, we're diving into the exciting world of sequences, specifically geometric sequences. We've got a sequence here, and our mission, should we choose to accept it (and we totally do!), is to figure out if it's geometric. If it is, then we'll become sequence sleuths and track down its first term and common ratio. Let's get started!

The Sequence in Question: $a_n = 5n^2 + 4$

Our suspect sequence is defined by the formula $a_n = 5n^2 + 4$. This formula tells us how to find any term in the sequence by plugging in the term number, 'n'. For example, to find the first term, we'd plug in n=1, for the second term, n=2, and so on. But before we start calculating terms, let's quickly recap what makes a sequence geometric.

What Makes a Sequence Geometric?

In the simplest terms, a geometric sequence is a sequence where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio. Think of it like a snowball rolling down a hill – it gets bigger and bigger by the same factor with each rotation. Mathematically, if we divide any term by its preceding term, we should always get the same common ratio if the sequence is geometric. This is a critical property we'll use to determine if our sequence fits the bill.

For a sequence to be definitively geometric, the ratio between consecutive terms must be constant. This constant ratio is the lifeblood of a geometric sequence; without it, the sequence simply doesn't qualify. So, how do we check for this consistency? We calculate the ratios between a few pairs of consecutive terms. If these ratios turn out to be the same, we're on the right track. However, if even one pair deviates from the common ratio, we know the sequence is not geometric. This is a crucial step in our investigation, ensuring we don't jump to conclusions based on just a few terms.

Therefore, before we even consider diving into the specifics of our sequence, we must firmly grasp the geometric sequence definition. It isn't merely about recognizing a pattern; it's about confirming a consistent multiplicative relationship between successive terms. This understanding is our compass, guiding us through the process of determining whether our sequence is geometric and, if so, what its defining characteristics are. So, with this principle firmly in mind, let's begin our analysis by calculating the first few terms and examining their relationships.

Let's Calculate Some Terms

Okay, let's put our formula to work and calculate the first few terms of our sequence. This will give us some concrete numbers to play with and help us spot any patterns (or lack thereof). Remember, $a_n = 5n^2 + 4$.

• For the first term (n=1): $a_1 = 5(1)^2 + 4 = 5 + 4 = 9$
• For the second term (n=2): $a_2 = 5(2)^2 + 4 = 5(4) + 4 = 20 + 4 = 24$
• For the third term (n=3): $a_3 = 5(3)^2 + 4 = 5(9) + 4 = 45 + 4 = 49$
• For the fourth term (n=4): $a_4 = 5(4)^2 + 4 = 5(16) + 4 = 80 + 4 = 84

So, our sequence starts like this: 9, 24, 49, 84, ... Now, let's see if these numbers behave like a geometric sequence.

To ascertain whether our sequence is geometric, we must meticulously examine the relationships between successive terms. This involves calculating the ratios between consecutive pairs, a process that will reveal whether a consistent multiplicative factor exists. The first step is to divide the second term by the first, thus establishing a potential ratio. We then proceed to divide the third term by the second, and the fourth by the third, and so on. Each of these calculations provides a snapshot of the relationship between terms, and it is the consistency of these snapshots that ultimately determines the sequence's nature.

If, after calculating several ratios, we find that they converge on a single value, we have strong evidence that the sequence is indeed geometric. This common value is not just a number; it's the common ratio that defines the sequence's growth pattern. However, if the ratios diverge, even slightly, we must conclude that the sequence is not geometric. This process of calculating and comparing ratios is therefore the linchpin of our analysis, guiding us toward an accurate determination of the sequence's properties. So, with our terms calculated, let's roll up our sleeves and delve into the ratios to uncover the truth about our sequence.

Checking for a Common Ratio

This is the moment of truth! To check for a common ratio, we'll divide each term by the term that comes before it. If we get the same result every time, we've got ourselves a geometric sequence. If not, well, our sequence is playing by different rules.

• Ratio between the second and first terms: $a_2 / a_1 = 24 / 9 = 8 / 3$ (approximately 2.67)
• Ratio between the third and second terms: $a_3 / a_2 = 49 / 24$ (approximately 2.04)

Uh oh! It looks like our ratios aren't the same. 8/3 is definitely not equal to 49/24. This means there's no common ratio, and our sequence isn't geometric.

Our investigation has led us to a critical juncture: the calculated ratios between consecutive terms are not uniform. This discovery is significant because it directly contradicts the fundamental requirement of a geometric sequence – the existence of a constant ratio. In a geometric sequence, each term is derived from its predecessor by multiplying by this common ratio, creating a predictable and consistent pattern of growth or decay. The lack of this consistency in our sequence's ratios signifies that the terms do not follow a simple multiplicative progression.

The implications of this finding are profound. It means that the formula $a_n = 5n^2 + 4$ generates a sequence that, while following a pattern, does so in a non-geometric manner. The terms increase, but not by a constant multiplicative factor. This realization helps us understand the broader landscape of sequences; not all sequences that exhibit a pattern are geometric, and discerning the specific type of pattern is crucial for further analysis. Therefore, with the ratios differing, we can definitively conclude that our sequence does not belong to the geometric family.

The Verdict

Based on our calculations, the sequence defined by $a_n = 5n^2 + 4$ is not geometric. There's no common ratio lurking here. So, we don't need to worry about finding the first term and common ratio – the sequence simply doesn't fit the criteria. We can confidently select the answer choice that indicates the sequence is not geometric.

Our exploration has reached its conclusion, and the results are clear: the sequence we examined does not exhibit the defining characteristics of a geometric progression. This determination is not merely an end point; it's a valuable learning opportunity. By identifying that this sequence is not geometric, we reinforce our understanding of what geometric sequences are and how they behave. It's like learning to distinguish between different species in the animal kingdom – each has its unique traits, and recognizing these traits allows us to classify them accurately. In this case, we've honed our ability to recognize the 'geometric species' of sequences, understanding that a constant ratio is the key identifier.

Moreover, this exercise highlights the importance of rigorous analysis. We didn't just assume the sequence was geometric based on a quick glance; we calculated ratios, compared them, and only then reached a conclusion. This methodical approach is crucial in mathematics, where precision and evidence-based reasoning are paramount. So, we not only solved a problem but also reinforced a valuable problem-solving skill. As we move forward, let's carry this understanding and analytical approach with us, ready to tackle new sequence challenges.

Final Answer

A. The sequence is not geometric.

And that's a wrap, folks! We've successfully determined that the sequence $a_n = 5n^2 + 4$ isn't geometric. Until next time, keep exploring the fascinating world of math!