Find A_{10} In The Sequence: Step-by-Step Solution

Hey guys! Let's dive into a common type of math problem you might encounter: finding a specific term in an arithmetic sequence. In this article, we'll break down a problem where we're given the formula for a sequence and need to find the value of the 10th term. Specifically, we'll tackle the sequence defined by $a_n = -3 + (n-1)9$. So, grab your thinking caps, and let's get started!

Understanding Arithmetic Sequences

Before we jump into solving for $a_{10}$, let's quickly recap what an arithmetic sequence actually 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. Think of it like a staircase where each step has the same height. The formula for the nth term ($a_n$) of an arithmetic sequence is usually given in the form:

$a_n = a_1 + (n-1)d$

Where:

• $a_n$ is the nth term we want to find.
• $a_1$ is the first term of the sequence.
• n is the position of the term in the sequence (e.g., 1 for the first term, 2 for the second term, and so on).
• d is the common difference.

In our case, we're given a slightly different form of the formula, but the principle is the same. Identifying the components and understanding their roles is key to successfully solving these problems. So let's break down our specific formula: $a_n = -3 + (n-1)9$.

Breaking Down the Given Formula: $a_n = -3 + (n-1)9$

Our formula, $a_n = -3 + (n-1)9$, might look a bit intimidating at first, but let's dissect it. We can see it's in a similar form to the general arithmetic sequence formula. Here's how it translates:

• The "-3" part corresponds to a starting point. It's not directly the first term ($a_1$) because of the way the formula is structured, but it's a related value. Think of it as an offset.
• The "(n-1)" part is the same as in the general formula, indicating that we're dealing with the position of the term we want to find.
• The "9" is the crucial piece – it's our common difference (d). This tells us that each term in the sequence is 9 more than the previous term.

Understanding these components is vital because it allows us to visualize how the sequence progresses. We start with something related to -3, and then we add 9 repeatedly to get the subsequent terms. Now that we've deciphered the formula, we're ready to tackle the main question: what is $a_{10}$?

Calculating $a_{10}$: Finding the 10th Term

Now for the fun part: actually calculating the value of $a_{10}$! This is where our formula becomes our best friend. We know we want to find the 10th term, so that means n = 10. We simply substitute n = 10 into our formula:

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

Now, let's simplify step-by-step, following the order of operations (PEMDAS/BODMAS):

1. Parentheses/Brackets: (10 - 1) = 9

   $a_{10} = -3 + (9)9$

2. Multiplication: 9 * 9 = 81

   $a_{10} = -3 + 81$

3. Addition: -3 + 81 = 78

So, there you have it! We've found that $a_{10} = 78$. That means the 10th term in the sequence is 78. It's really that straightforward! The key is to carefully substitute the value of n and then follow the order of operations to simplify the expression.

Verifying the Result and Exploring the Sequence

It's always a good idea to double-check your work, especially in math. One way to verify our result is to think about the sequence conceptually. We know the common difference is 9. We can manually calculate the first few terms to get a feel for the sequence:

• $a_1 = -3 + (1-1)9 = -3 + 0 = -3$
• $a_2 = -3 + (2-1)9 = -3 + 9 = 6$
• $a_3 = -3 + (3-1)9 = -3 + 18 = 15$
• $a_4 = -3 + (4-1)9 = -3 + 27 = 24$

And so on... We can see the sequence is increasing by 9 each time: -3, 6, 15, 24... If we continued this pattern, we would eventually reach 78 as the 10th term. This gives us confidence that our calculation is correct. Furthermore, this exercise highlights the beauty of arithmetic sequences – their predictable and consistent nature makes them relatively easy to analyze and work with. Understanding the underlying pattern allows us to solve for any term in the sequence without having to calculate all the preceding terms.

Common Mistakes to Avoid

When working with arithmetic sequences, there are a few common pitfalls to watch out for. Avoiding these mistakes can save you time and frustration:

• Incorrect Order of Operations: As we emphasized earlier, following the order of operations (PEMDAS/BODMAS) is crucial. Make sure you perform the operations in the correct sequence (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). A simple mistake here can lead to a completely wrong answer.
• Misinterpreting the Formula: It's essential to correctly identify the components of the formula, especially the common difference (d) and how they relate to the term number (n). Confusing these can lead to incorrect substitution and calculations.
• Arithmetic Errors: Simple addition, subtraction, multiplication, or division errors can happen, especially when dealing with larger numbers. Double-check your calculations to minimize these mistakes. Using a calculator can be helpful, but make sure you understand the steps you're performing.
• Forgetting the (n-1) Term: The (n-1) term in the formula is vital. Forgetting to subtract 1 from n before multiplying by the common difference will result in an incorrect answer. Remember that we're looking at the difference relative to the first term, so we need to account for that offset.

By being aware of these common mistakes, you can significantly improve your accuracy and confidence when working with arithmetic sequences.

Practice Problems: Test Your Understanding

Now that we've walked through the solution and discussed potential pitfalls, it's time to put your knowledge to the test! Here are a couple of practice problems for you to try:

1. Given the arithmetic sequence $a_n = 5 + (n-1)4$, find the value of $a_{15}$.
2. An arithmetic sequence is defined by $a_n = -10 + (n-1)3$. What is the value of $a_{20}$?

Try solving these problems on your own, using the steps we outlined earlier. Don't be afraid to refer back to the article if you need a reminder. The key to mastering any mathematical concept is practice, practice, practice! The more problems you solve, the more comfortable and confident you'll become.

Conclusion: Mastering Arithmetic Sequences

So, guys, we've successfully navigated the world of arithmetic sequences and learned how to find a specific term given the formula. We broke down the formula, identified the key components, and carefully calculated the 10th term ($a_{10}$) in our example sequence. Remember, the key takeaways are:

• Understand the formula for an arithmetic sequence: $a_n = a_1 + (n-1)d$ (or variations of it).
• Identify the common difference (d) and its role in the sequence.
• Substitute the correct value for n (the term number).
• Follow the order of operations (PEMDAS/BODMAS) carefully.
• Practice, practice, practice!

By mastering these concepts, you'll be well-equipped to tackle a wide range of problems involving arithmetic sequences. Keep practicing, and you'll become a math whiz in no time! Good luck, and keep learning!