Finding The Fifth Term Of An Arithmetic Sequence

by Andrew McMorgan 49 views

Hey there, math enthusiasts! Let's dive into a cool problem involving arithmetic sequences. We'll break down how to find the fifth term of a sequence, given its first term and common difference. This is a fundamental concept, and understanding it will definitely boost your math game. So, let's get started!

Understanding the Basics: Arithmetic Sequences

Alright, before we jump into the problem, let's make sure we're all on the same page about arithmetic sequences. Imagine a sequence of numbers where the difference between any two consecutive terms is constant. That constant difference is what we call the common difference. Pretty straightforward, right? Think of it like climbing stairs – each step (the difference) is the same height.

Formally, an arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. The general form of an arithmetic sequence is:

  • a, a + d, a + 2d, a + 3d, ...

where:

  • 'a' is the first term
  • 'd' is the common difference

In our case, we're given that the first term is 8 (a = 8), and the common difference is k (d = k). Our goal is to find the fifth term of this sequence. To find the nth term of an arithmetic sequence, we use the following formula:

  • an = a + (n - 1) * d

Where:

  • an is the nth term
  • a is the first term
  • n is the term number
  • d is the common difference

For example, if the first term is 2 and the common difference is 3, the sequence would look like this: 2, 5, 8, 11, and so on. Understanding this foundational concept is key to solving the problem, and helps you work through other problems. Remember, the difference between consecutive terms is always the same. This is the hallmark of an arithmetic sequence. Keep this in mind, and you'll be golden.

Problem Breakdown: Identifying the Key Information

Now, let's get back to our problem. We're given that k is a positive real number. This means k can be any positive value, like 1, 2.5, or even pi. The first term of our arithmetic sequence is 8, and the common difference is k. We want to find the fifth term. Let's list what we know:

  • First term (a) = 8
  • Common difference (d) = k
  • We want to find the fifth term, which means n = 5

See? It's all about breaking down the problem into smaller, manageable pieces. Knowing what we have and what we need to find makes the solution much clearer. By taking our time and listing out the given information, we've set ourselves up for success. Understanding the key parameters of the problem is essential. Make sure you're crystal clear on the first term, common difference, and which term you're trying to find. This approach helps prevent errors and makes the problem easier to solve. You’ll be surprised how much easier things become once you break them down.

Solving for the Fifth Term

We've got the necessary ingredients, so let's cook up the solution! We know the formula for the nth term of an arithmetic sequence is: an = a + (n - 1) * d.

We need to find the fifth term (a5), so we'll plug in the values we have:

  • a5 = 8 + (5 - 1) * k

Simplify the equation:

  • a5 = 8 + 4 * k

So, the fifth term of the arithmetic sequence is 8 + 4*k. Easy peasy, right?

This simple formula is your best friend when dealing with arithmetic sequences. Remember to pay close attention to the order of operations – parentheses first! Once you plug in the values, it's all about careful calculations. Making sure you understand each step is crucial. This helps reinforce the underlying concepts. Practice different examples, and you'll become a pro in no time! The fifth term is essentially the first term plus four times the common difference.

Matching the Answer to the Options

Now that we've found our answer (8 + 4*k), let's compare it to the answer choices provided. Remember, the goal here is to make sure your solution aligns with one of the given options. After carefully examining the options given, we should find that the answer is:

  • D) 8 + 4k

Congratulations! You've successfully solved the problem. You took the information, used the formula, and arrived at the correct answer. Give yourself a pat on the back.

Conclusion: Mastering Arithmetic Sequences

So, there you have it, guys! We've successfully navigated through the problem of finding the fifth term of an arithmetic sequence. By understanding the concept of arithmetic sequences, the role of common differences, and the formula to find the nth term, we were able to solve it.

Remember, practice is key! Try working through similar problems on your own. You can change the first term, the common difference, and the term you want to find. The more you practice, the more comfortable you'll become with this concept. You'll become proficient and confident in no time. This skill is useful for other areas of math and real-world applications. Keep exploring, keep learning, and don't be afraid to tackle new challenges. You've got this!