Find The 10th Term Of An Arithmetic Sequence

Hey math lovers! Ever stumbled upon a sequence and wondered what a specific term would be way down the line? Today, we're diving deep into the world of arithmetic sequences, and I'm gonna show you guys exactly how to nail down that elusive 10th term using the formula $a_n=1+\frac{1}{3}(n-1)$. This isn't just about crunching numbers; it's about understanding the pattern that makes sequences tick. So grab your notebooks, maybe a snack, and let's get this party started!

Understanding Arithmetic Sequences: The Basics

Alright, let's talk about what an arithmetic sequence actually is. Think of it as a list of numbers where the difference between any two consecutive terms is always the same. This constant difference is called the "common difference," and it's the secret sauce that keeps the sequence marching forward in a predictable way. Our formula, $a_n = 1 + \frac{1}{3}(n-1)$, is a prime example of how we define an arithmetic sequence. Here, '$a_n$' represents the term at any given position '$n$' in the sequence. The '1' at the beginning is our first term (when $n=1$, $a_1 = 1 + \frac{1}{3}(1-1) = 1$). The '$\frac{1}{3}$' is our common difference – every time we move to the next term, we add this value. And '$n-1$'? That's just a handy way to make sure we're applying the common difference the correct number of times to get to the '$n$th' term.

So, why is this formula so cool? Because it gives us a direct ticket to any term in the sequence without having to list out every single number before it. Imagine if you needed to find the 100th term – writing out 99 numbers would be a drag, right? But with this formula, it's a piece of cake. We just plug in the desired term number for '$n$' and boom, there it is! This formula is the bedrock of solving many sequence problems, and understanding its components—the first term, the common difference, and the term number '$n$'—is crucial. It's like knowing the ingredients to bake the perfect cake. The first term is your foundation, the common difference is the flavor, and '$n$' is how much cake you want!

Decoding the Formula: $a_n = 1 + \frac{1}{3}(n-1)$

Let's break down this specific formula, $a_n = 1 + \frac{1}{3}(n-1)$, like we're dissecting a cool gadget. First off, the '$a_n$' is your target – it's the value of the term you want to find. The '1' right after the equals sign? That's our first term, $a_1$. It's the starting point of our sequence. If you were to plug in $n=1$ into the formula, you'd get $a_1 = 1 + \frac{1}{3}(1-1) = 1 + \frac{1}{3}(0) = 1$. See? It checks out. Now, the common difference is the number being multiplied by $(n-1)$, which is $\frac{1}{3}$. This means that to get from one term to the next, we consistently add $\frac{1}{3}$. For example, the second term ($n=2$) would be $a_2 = 1 + \frac{1}{3}(2-1) = 1 + \frac{1}{3}(1) = 1 \frac{1}{3}$. The third term ($n=3$) would be $a_3 = 1 + \frac{1}{3}(3-1) = 1 + \frac{1}{3}(2) = 1 + \frac{2}{3} = 1 \frac{2}{3}$. You can see the pattern: we add $\frac{1}{3}$ each time. The $(n-1)$ part is super important because it tells us how many times the common difference has been added to the first term to reach the $n$th term. For the first term ($n=1$), we add the difference 0 times. For the second term ($n=2$), we add it 1 time. For the third term ($n=3$), we add it 2 times, and so on. It's like taking steps – you need $n-1$ steps to get to the $n$th position starting from the first position. This formula is elegant because it encapsulates all the necessary information to generate any term in the sequence. It's a concise representation of the sequence's rule.

Understanding these components allows us to predict the behavior of the sequence. We know it starts at 1 and increases by $\frac{1}{3}$ for every subsequent term. This knowledge is fundamental not just for finding a specific term but also for grasping concepts like the sum of an arithmetic series or analyzing trends in data that follow a linear progression. So, when you see a formula like this, don't be intimidated! Just remember that each part has a specific role, and together they define the entire universe of that sequence. It’s like deciphering a secret code, and once you know the key, everything becomes clear. This formula is your key to unlocking the secrets of this particular arithmetic sequence, making it easy to find any term you desire with just a bit of substitution and calculation. The structure is logical, building from a starting point with incremental additions, making it a powerful tool for mathematical exploration and problem-solving in various contexts.

Calculating the 10th Term

Now for the main event, guys! We need to find the 10th term of the sequence defined by $a_n = 1 + \frac{1}{3}(n-1)$. Remember how we decoded the formula? The '$n$' in the formula represents the position of the term we're interested in. Since we want the 10th term, we simply need to substitute $n=10$ into our formula. So, we're looking for $a_{10}$.

Let's plug it in:

$a_{10} = 1 + \frac{1}{3}(10-1)$

First, we handle the part inside the parentheses: $10-1 = 9$.

So now our equation looks like this:

$a_{10} = 1 + \frac{1}{3}(9)$

Next, we perform the multiplication. We multiply $\frac{1}{3}$ by 9:

$\frac{1}{3} \times 9 = \frac{9}{3} = 3$

Finally, we add this result to the first term:

$a_{10} = 1 + 3$

$a_{10} = 4$

And there you have it! The 10th term of the arithmetic sequence defined by $a_n = 1 + \frac{1}{3}(n-1)$ is 4. It was that straightforward! By substituting the term number '$n$' into the formula, we can directly calculate the value of any term in the sequence. This method is efficient and accurate, saving us the trouble of calculating all the preceding terms.

Think about it: if we had to list them out, we'd have $a_1=1$, $a_2=1\frac{1}{3}$, $a_3=1\frac{2}{3}$, $a_4=2$, $a_5=2\frac{1}{3}$, $a_6=2\frac{2}{3}$, $a_7=3$, $a_8=3\frac{1}{3}$, $a_9=3\frac{2}{3}$, and finally $a_{10}=4$. See? It matches! Doing it this way is great for visualizing the sequence, but when you need a term far down the line, the formula is your best friend. It's a powerful shortcut that respects the underlying mathematical structure.

This process highlights the predictive power of mathematical formulas. Once the rule is established (the formula), any future state (any term) can be determined with certainty. This principle is fundamental in many areas of science and engineering, where models are used to predict outcomes based on known relationships. In this case, the arithmetic sequence formula provides a clear and simple model for generating terms. The substitution of $n=10$ is a direct application of this model, yielding the precise value for the 10th term. It demonstrates that with a clear definition and a specific input, a predictable and accurate output is guaranteed. This is the beauty of mathematics: consistency and reliability in its operations, allowing us to solve problems systematically and confidently.

Exploring the Options: Which Answer is Right?

So, we've calculated that the 10th term is 4. Now, let's look at the options provided to see which one matches our answer:

(A) 3 (B) 4 (C) $4 \frac{1}{3}$ (D) 12

Our calculated value is 4, which perfectly matches Option (B). High five for getting it right! This confirms our understanding and calculation using the arithmetic sequence formula. It’s always a good practice to double-check your work, especially when multiple-choice options are involved, to ensure you haven’t made any silly calculation errors.

Sometimes, students might mistakenly add the common difference one extra time, or forget to subtract 1 from $n$. For instance, if you forgot the $(n-1)$ and just used '$n$', you'd get $1 + \frac{1}{3}(10) = 1 + \frac{10}{3} = 1 + 3\frac{1}{3} = 4\frac{1}{3}$, which is option (C). This is a common pitfall, and it highlights the importance of correctly interpreting the formula. Another mistake could be using $n=9$ instead of $n=10$, which would give $1 + \frac{1}{3}(9-1) = 1 + \frac{1}{3}(8) = 1 + \frac{8}{3} = 1 + 2\frac{2}{3} = 3\frac{2}{3}$, not an option but close to (A). Or perhaps confusing the first term with the common difference, or vice versa. It's crucial to identify each component of the formula correctly: the initial value and the rate of change.

Understanding these potential errors can actually help reinforce the correct method. By recognizing why other answers might seem plausible but are incorrect, you solidify your grasp on the actual process. In this case, the $(n-1)$ factor is key. It adjusts the number of times the common difference is applied. Since we start at the first term, we only need to add the difference $n-1$ times to reach the $n$th term. This is a subtle but critical point in arithmetic sequence formulas. So, when you encounter such problems, remember to carefully identify the first term, the common difference, and the term number, and then apply the formula systematically. Always check your arithmetic, and if possible, see if the answer makes intuitive sense in the context of the sequence's progression.

Why other options are incorrect

Let's quickly debunk why the other options aren't the correct answer for the 10th term of $a_n=1+\frac{1}{3}(n-1)$:

• (A) 3: To get 3, you would need $1 + \frac{1}{3}(n-1) = 3$, which means $\frac{1}{3}(n-1) = 2$, so $n-1 = 6$, and $n=7$. So, 3 is the 7th term, not the 10th.
• (C) $4 \frac{1}{3}$: As discussed earlier, this answer comes from incorrectly using '$n$' instead of '$(n-1)$' in the formula: $1 + \frac{1}{3}(10) = 1 + \frac{10}{3} = 1 + 3\frac{1}{3} = 4\frac{1}{3}$. This is the value you'd get if you were adding the common difference $n$ times to the zeroth term (if that existed), or incorrectly applied the formula.
• (D) 12: This number is significantly larger than our calculated value. It's hard to arrive at 12 with a simple miscalculation from the given formula. Perhaps one might incorrectly multiply the common difference by $n$ and then add the first term, $1 + (\frac{1}{3} imes 10) = 4\frac{1}{3}$ or maybe $1 + (3 imes 10) = 31$ (if they confused the common difference with 3), or $1 + (1 imes 10) = 11$ (if they ignored the fraction). Getting 12 would likely involve a more complex error or a misunderstanding of the formula's structure.

It's always good to understand why the wrong answers are wrong. It helps solidify your understanding of the correct method and prevents you from falling into common traps. Stick to the formula, substitute carefully, and calculate step-by-step, and you'll be golden!

Conclusion: Mastering Arithmetic Sequences

So, there you have it, folks! We've successfully found the 10th term of the arithmetic sequence $a_n = 1 + \frac{1}{3}(n-1)$ by plugging in $n=10$ and following the formula step-by-step. The answer, as we found, is 4. This exercise underscores the power and simplicity of using formulas to define and navigate sequences. Whether you're dealing with arithmetic, geometric, or other types of sequences, understanding the defining formula is your key to unlocking any term, sum, or pattern within them.

Remember, arithmetic sequences are characterized by a constant difference between consecutive terms. The formula $a_n = a_1 + d(n-1)$ (where $a_1$ is the first term and $d$ is the common difference) is your universal toolkit. In our case, $a_1 = 1$ and $d = \frac{1}{3}$. By substituting $n=10$, we directly computed $a_{10} = 1 + \frac{1}{3}(10-1) = 1 + \frac{1}{3}(9) = 1 + 3 = 4$. It's a clean, direct calculation that bypasses the need to list out intermediate terms.

Keep practicing these types of problems, guys! The more you work with these formulas, the more intuitive they become. Understanding sequences is a fundamental skill in mathematics that opens doors to more advanced topics like series, calculus, and discrete mathematics. Plus, it's incredibly satisfying to be able to predict future values based on a given rule. So next time you see a sequence, don't just see a list of numbers; see a pattern, a rule, and a potential for prediction. Keep those math brains sharp and continue exploring the amazing world of numbers! Happy problem-solving!

• Keywords: arithmetic sequence, nth term, sequence formula, mathematics, common difference, first term
• Difficulty: Easy
• Learning Objectives: Understand the definition of an arithmetic sequence, apply the formula for the nth term, substitute values into a formula, perform calculations involving fractions and order of operations.
• Context: This problem is typical in introductory algebra or pre-calculus courses, focusing on the foundational concepts of sequences and series.
• Problem Breakdown: The problem requires identifying the components of the given arithmetic sequence formula and using them to calculate a specific term. It tests the ability to substitute a value for a variable and perform arithmetic operations correctly. The multiple-choice format also requires careful verification of the result against the given options, checking for common errors.

Final Answer:

The final answer is (B) 4.

This is derived directly from the formula $a_n=1+\frac{1}{3}(n-1)$ by substituting $n=10$: $a_{10}=1+\frac{1}{3}(10-1)=1+\frac{1}{3}(9)=1+3=4$.