Solving For X: Arithmetic Progressions And Integer Solutions

Hey guys! Let's dive into a fun math problem. We're going to figure out the value of x. The cool part? We're dealing with something called an arithmetic progression (AP). Basically, this means we have a sequence of numbers where the difference between each consecutive term is the same. Think of it like a staircase; each step is the same height.

So, what's our problem? We're given three terms: 11 - x, 2x - 1, and 9x + 3. We're told these terms form an AP. And, to make things even more interesting, we know that x is an integer. That means x can be a whole number, like 1, 2, -5, or 0. No fractions or decimals allowed! Our goal is to use the properties of arithmetic progressions to find the specific integer value of x that makes this sequence work. We will break down the problem step by step to find the value of x.

Now, let's talk about the key property of an AP. The difference between the second term and the first term is the same as the difference between the third term and the second term. In math terms: (second term - first term) = (third term - second term). It's all about that constant difference, you see? This consistent difference is what defines the arithmetic progression.

To solve this, we'll set up an equation using this property. We will substitute the given terms into the equation and then simplify it to isolate x. From here, we will solve the equation and confirm that we are getting an integer solution. Finally, we can check our answer by plugging the value of x back into our original terms to ensure they form a valid AP. Ready to get started? Let's get down to business and find that x!

Understanding Arithmetic Progressions (AP)

Alright, let's make sure we're all on the same page about arithmetic progressions. An arithmetic progression (AP), as we mentioned before, is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is often called the 'common difference,' and it's super important. Let's say we have an AP with the terms a1, a2, a3, and so on. The common difference (d) is calculated as: d = a2 - a1 = a3 - a2 = and so on. It's the same difference throughout the entire sequence. Understanding this is key to solving our problem.

Let's imagine some examples to make it super clear. The sequence 2, 4, 6, 8, 10 is an AP. The common difference is 2 (4 - 2 = 2, 6 - 4 = 2, etc.). Another example: 10, 7, 4, 1, -2 is also an AP. Here, the common difference is -3 (7 - 10 = -3, 4 - 7 = -3, etc.). Notice how the common difference can be positive, negative, or even zero if all the terms are the same. This concept of the common difference allows us to predict the next term in the sequence. If we know the first term and the common difference, we can find any term in the AP.

Knowing the common difference also helps us to verify if a sequence is an AP. If the differences between consecutive terms aren't constant, then the sequence isn't an AP. For example, the sequence 1, 2, 4, 7, 11 is not an AP because the differences (1, 2, 3, and 4) aren't the same. The consistent difference is the characteristic feature of an AP. The ability to recognize an AP and calculate its common difference unlocks many possibilities in mathematical problems.

In our problem, the three terms 11 - x, 2x - 1, and 9x + 3 must form an AP. So, the difference between 2x - 1 and 11 - x has to equal the difference between 9x + 3 and 2x - 1. That's the fundamental principle we will use to find x. Now, with this understanding of what is an AP, we are ready to find the x.

Setting up the Equation

Okay, guys, it's time to translate our understanding of APs into a concrete equation. Remember, for an AP, the difference between consecutive terms is constant. In our case, this means:

(2x - 1) - (11 - x) = (9x + 3) - (2x - 1)

This equation is the heart of our problem. It states that the difference between the second term and the first term is the same as the difference between the third term and the second term. Each term is in its correct place and we can now simplify each side of the equation. This setup lets us create a straightforward path to solving for x.

Let's break down the simplification process step by step, making it easy to follow: First, we need to carefully expand both sides of the equation. Pay close attention to the signs, especially when subtracting expressions.

On the left side: (2x - 1) - (11 - x) becomes 2x - 1 - 11 + x. On the right side: (9x + 3) - (2x - 1) becomes 9x + 3 - 2x + 1. Next, we combine like terms on each side. On the left side, 2x + x = 3x, and -1 - 11 = -12. So the left side simplifies to 3x - 12. On the right side, 9x - 2x = 7x, and 3 + 1 = 4. So the right side simplifies to 7x + 4. Now, the simplified equation is: 3x - 12 = 7x + 4.

This simplified equation is much easier to work with, right? The expansion and simplification are crucial steps in solving the equation. Once we have a simplified equation, isolating x will be easier.

Solving for X

Alright, we've got our simplified equation: 3x - 12 = 7x + 4. Now it's time to isolate x and find its value. Remember, our goal is to get x alone on one side of the equation. Here’s how we'll do it.

First, let's get all the x terms on one side. We can subtract 3x from both sides to eliminate it from the left side: 3x - 12 - 3x = 7x + 4 - 3x. This simplifies to -12 = 4x + 4. Next, we need to get rid of the constant term on the right side. We can subtract 4 from both sides: -12 - 4 = 4x + 4 - 4. This simplifies to -16 = 4x. Finally, to isolate x, we divide both sides by 4: -16 / 4 = 4x / 4. This gives us x = -4.

So, we've found that x = -4. But we're not done yet! We need to make sure this is actually a valid solution and it's an integer, as the problem specified. Now we can check to verify. Let's do that in the next section.

Verifying the Solution

Great job on getting that x value! But as any good mathematician knows, we need to check our work. It's time to plug our value for x back into the original terms and make sure they indeed form an AP.

We found that x = -4. Let's substitute this value into each of the original terms: The first term was 11 - x. Substituting x = -4, we get 11 - (-4) = 11 + 4 = 15. The second term was 2x - 1. Substituting x = -4, we get 2(-4) - 1 = -8 - 1 = -9. The third term was 9x + 3. Substituting x = -4, we get 9(-4) + 3 = -36 + 3 = -33.

So, the three terms, when x = -4, are 15, -9, and -33. Now, let's verify if they form an AP. The difference between the second and first terms is -9 - 15 = -24. The difference between the third and second terms is -33 - (-9) = -33 + 9 = -24. The common difference is consistent! Since the common difference is constant, the terms form an AP. Thus, our solution is correct!

Also, since -4 is an integer, our solution meets all the problem requirements. We found the value of x (-4) and verified that the resulting terms form an AP with an integer value for x. Success!

Conclusion

Awesome work, everyone! We successfully solved for x in the arithmetic progression problem. We went through a few crucial steps. First, we understood the properties of an arithmetic progression. Next, we set up an equation, simplified it, and carefully solved for x. Finally, we verified our solution by plugging x back into the original terms to confirm they formed a valid AP. And x turned out to be -4, a perfect integer solution! This problem highlighted the importance of understanding the concepts behind the equation and the need for careful calculations and verification. Understanding APs is a key skill. Keep practicing, and you'll become a math whiz in no time!

I hope you enjoyed the explanation. Keep practicing and exploring math. If you have any questions, feel free to ask! Thanks for joining me in this math adventure, and I hope to see you again for the next one!