Nth Term Of Arithmetic Sequence: Formula & Examples

Have you ever wondered how to predict the next number in a sequence? Or maybe you're curious about finding a specific term, like the 100th, without listing out all the numbers before it? That's where the concept of the nth term in an arithmetic sequence comes in handy! Let's dive into the fascinating world of sequences and learn how to find the expression for the nth term, using the sequence 11, 6, 1, -4, -9, ... as our example.

Understanding Arithmetic Sequences

Before we jump into finding the nth term, let's make sure we're all on the same page about what an arithmetic sequence is. An arithmetic sequence is simply a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'. In simpler terms, you're adding (or subtracting) the same number each time to get the next term.

In our example sequence, 11, 6, 1, -4, -9, ..., we can see that the difference between consecutive terms is consistently -5 (6 - 11 = -5, 1 - 6 = -5, and so on). So, this is indeed an arithmetic sequence with a common difference (d) of -5. Spotting this common difference is the crucial first step in unraveling the mystery of the nth term.

Recognizing patterns is a foundational skill in mathematics, and arithmetic sequences provide a clear and structured environment to hone this ability. Consider other sequences: 2, 4, 6, 8... (d = 2) or 1, 5, 9, 13... (d = 4). Each presents a unique 'stepping stone' defined by its common difference. Understanding this constant progression allows us to move beyond simply observing the sequence and venture into predicting its future terms. The nth term formula, which we'll explore shortly, is the key that unlocks this predictive power. So, always remember to first identify that common difference – it's the heartbeat of the arithmetic sequence.

The Formula for the nth Term

Now that we know what an arithmetic sequence is, let's introduce the formula that will help us find the nth term. The formula looks like this:

an = a1 + (n - 1)d

Where:

• an is the nth term (the term we want to find)
• a1 is the first term of the sequence
• n is the position of the term we want to find (e.g., 1st term, 10th term, 100th term)
• d is the common difference

This formula might seem a bit intimidating at first glance, but it's actually quite straightforward once you understand each component. The beauty of this formula lies in its ability to connect any term in the sequence to its position, the starting point (first term), and the consistent 'step' (common difference). Think of it as a roadmap where you start at a specific location (a1), take a certain number of steps (n - 1) of a fixed size (d), and arrive at your destination (an). Let’s break down each part again to solidify our understanding. 'an' is the term you are seeking – it’s the unknown you're trying to uncover. 'a1' is your known starting point, the first term in the sequence. 'n' is the term number; if you want the 5th term, n is 5. And 'd', as we’ve discussed, is that constant difference that defines the sequence's rhythm. Mastering this formula opens doors to solving a myriad of problems related to arithmetic sequences.

Applying the Formula to Our Example

Let's use the formula to find the expression for the nth term of our sequence: 11, 6, 1, -4, -9, ... We already know:

• a1 = 11 (the first term)
• d = -5 (the common difference)

Now, we simply plug these values into the formula:

an = 11 + (n - 1)(-5)

This is a great start, but we can simplify this expression further to make it more user-friendly. Remember, our goal is to find a clear and concise representation of the nth term.

Stepping through the substitution is crucial for understanding how the formula translates to a specific sequence. We're essentially mapping the abstract formula onto the concrete data of our sequence. But stopping at the substitution is like having a map and knowing the starting point but not charting the route to the destination. The simplification step is where the magic happens. It transforms the raw equation into an elegant expression that directly links the term number 'n' to the value of the term. By simplifying, we reveal the underlying pattern of the sequence in its most accessible form. This makes it not only easier to calculate any term but also to grasp the sequence’s behavior as 'n' changes. So, let’s not leave our map half-drawn; let's simplify and uncover the complete route.

Simplifying the Expression

To simplify the expression, we need to distribute the -5 and combine like terms:

an = 11 + (n - 1)(-5)
an = 11 - 5n + 5
an = -5n + 16

So, the expression for the nth term of the sequence 11, 6, 1, -4, -9, ... is an = -5n + 16.

This simplified form is powerful because it directly tells us how to find any term in the sequence. If you want to find the 10th term, simply substitute n = 10 into the formula: a10 = -5(10) + 16 = -34. This is much easier than manually extending the sequence! The algebraic manipulation, going from the initial substitution to the simplified form, is a testament to the power of algebra in revealing underlying structure. Each step, distributing, combining like terms, is a deliberate act of shaping the equation into its most insightful form. The result, an = -5n + 16, is not just an answer; it's a compact model of the entire sequence. It encapsulates the sequence's essence, allowing us to predict terms far down the line with ease.

Testing the Formula

To make sure our formula is correct, we can test it with a few terms from the original sequence. Let's try n = 1, n = 2, and n = 3:

• For n = 1: a1 = -5(1) + 16 = 11 (Correct!)
• For n = 2: a2 = -5(2) + 16 = 6 (Correct!)
• For n = 3: a3 = -5(3) + 16 = 1 (Correct!)

Since the formula works for these terms, we can be confident that it is the correct expression for the nth term.

This verification step is not just a formality; it's a crucial safeguard against errors. Think of it as proofreading your work before submitting it. By plugging in known values, we're essentially asking the formula to 'show its work.' If it consistently produces the correct terms, it earns our trust. If it stumbles, it signals a need to revisit our calculations. Testing also deepens our understanding of the formula's behavior. We see how the value of 'n' directly influences the result, reinforcing the connection between the term number and the term value. This process not only validates our answer but also solidifies our grasp of the underlying concepts. So, always make testing a non-negotiable part of your problem-solving routine.

Conclusion

Finding the expression for the nth term of an arithmetic sequence is a fundamental skill in mathematics. By understanding the formula an = a1 + (n - 1)d and practicing its application, you can confidently tackle any arithmetic sequence problem. Remember to identify the first term (a1) and the common difference (d), plug them into the formula, simplify the expression, and always test your result to ensure accuracy. With this knowledge, you're well-equipped to explore the fascinating world of sequences and patterns!

Understanding arithmetic sequences and their nth terms is more than just memorizing a formula; it's about developing a mathematical way of thinking. It's about recognizing patterns, formulating rules, and using those rules to make predictions. This skill is applicable far beyond the realm of mathematics, in fields ranging from computer science to finance. The ability to identify trends and extrapolate future behavior is a valuable asset in any discipline. So, mastering the nth term formula is not just about solving a specific type of problem; it's about cultivating a powerful problem-solving mindset. Embrace the challenge, practice consistently, and you'll find that the world of mathematics, like arithmetic sequences themselves, unfolds with elegant predictability.

For further exploration of arithmetic sequences and related concepts, you might find the resources available at Khan Academy's Arithmetic Sequences Section to be highly beneficial. 🚀