Finding P Value In A Linear Sequence A Math Problem

Hey guys! Ever wondered how to find a missing piece in a sequence? Let's dive into a cool math problem where we'll figure out the value of 'p' that makes a sequence linear. It's like solving a puzzle, and trust me, it's super satisfying when you crack it!

Understanding Linear Sequences

Before we jump into the problem, let's quickly recap what a linear sequence is. Think of it as a sequence where the difference between consecutive terms is always the same. It's like climbing stairs where each step is the same height. This constant difference is often called the common difference.

For example, in the sequence 2, 4, 6, 8..., the common difference is 2. Each term is obtained by adding 2 to the previous term. Simple, right? This consistent pattern is the key to solving our problem.

Now, let's consider another example to solidify this concept. Take the sequence 1, 5, 9, 13... Here, the common difference is 4. Notice how each number increases by 4? This consistency is what defines a linear sequence. Understanding this basic principle is crucial because it allows us to predict future terms in the sequence and, more importantly, helps us find missing values like 'p' in our problem. We're essentially looking for a value of 'p' that maintains this constant difference between the given terms. So, with this understanding, let's tackle the main problem and see how we can apply this to find the value of 'p'.

Setting Up the Equation

So, our mission is to find the value of p that makes the terms 2p - 1, 5p + 3, and 11 form a linear sequence. Remember the golden rule of linear sequences: the difference between consecutive terms is constant. This is our secret weapon!

To put this into action, we can set up an equation. The difference between the second term (5p + 3) and the first term (2p - 1) should be equal to the difference between the third term (11) and the second term (5p + 3). Mathematically, this looks like:

(5p + 3) - (2p - 1) = 11 - (5p + 3)

This equation is the backbone of our solution. It directly uses the property of constant difference in linear sequences. By setting up the equation this way, we're essentially saying that the 'step' from the first term to the second term is the same size as the 'step' from the second term to the third term. This equation is our roadmap, guiding us towards the value of p that satisfies the condition of a linear sequence. Now, our next step is to simplify and solve this equation. We'll use our algebra skills to isolate p and find its value. Let’s get to it!

Solving for p

Alright, let's roll up our sleeves and solve the equation we set up:

(5p + 3) - (2p - 1) = 11 - (5p + 3)

First, we need to simplify both sides of the equation. Let’s tackle the left side first. Distribute the negative sign in the second term:

5p + 3 - 2p + 1 = 11 - (5p + 3)

Now, combine like terms on the left side:

3p + 4 = 11 - (5p + 3)

Next, let's simplify the right side. Again, distribute the negative sign:

3p + 4 = 11 - 5p - 3

Combine like terms on the right side:

3p + 4 = 8 - 5p

Now, our equation looks much cleaner. It's time to isolate p. Let's add 5p to both sides:

3p + 5p + 4 = 8 - 5p + 5p

This simplifies to:

8p + 4 = 8

Next, subtract 4 from both sides:

8p + 4 - 4 = 8 - 4

Which gives us:

8p = 4

Finally, to solve for p, divide both sides by 8:

p = 4 / 8

Simplify the fraction:

p = 1 / 2

Ta-da! We've found that p equals 1/2. This means that when p is 1/2, the terms 2p - 1, 5p + 3, and 11 will form a linear sequence. But hold on, we're not quite done yet. It's always a good idea to check our answer to make sure it fits the original problem. Let's plug p = 1/2 back into the terms and see if they form a linear sequence.

Verification

We've found that p = 1/2, but let's make sure our answer is correct. To do this, we'll substitute p = 1/2 back into the original terms of the sequence: 2p - 1, 5p + 3, and 11. This is like the final piece of the puzzle, ensuring everything fits perfectly.

First, let's calculate the first term:

2p - 1 = 2*(1/2) - 1 = 1 - 1 = 0

So, the first term is 0.

Next, let's find the second term:

5p + 3 = 5*(1/2) + 3 = 2.5 + 3 = 5.5

The second term is 5.5.

The third term is already given as 11.

Now, our sequence looks like this: 0, 5.5, 11. To verify if this is a linear sequence, we need to check if the difference between consecutive terms is constant.

Let's find the difference between the second and first terms:

5. 5 - 0 = 5.5

Now, let's find the difference between the third and second terms:

6. • 5.5 = 5.5

Great! The difference between consecutive terms is the same (5.5). This confirms that the sequence 0, 5.5, 11 is indeed a linear sequence. Therefore, our value of p = 1/2 is correct!

So, there you have it! We successfully found the value of p that makes the given terms a linear sequence. This problem beautifully illustrates how understanding the properties of sequences and using algebraic techniques can help us solve mathematical puzzles. Remember, the key is to break down the problem into smaller, manageable steps and use the information given to guide your solution.

Conclusion

Isn't math cool? We started with a sequence with a missing piece and, using the properties of linear sequences and some algebra magic, we found the value of p. Remember, the trick is to understand the core concept – in this case, the constant difference in a linear sequence – and then apply it to set up an equation. Solving for p was like unlocking a secret code!

This kind of problem-solving is not just about getting the right answer; it's about developing your logical thinking and problem-solving skills. These skills are super valuable, not just in math class, but in all aspects of life. So, keep practicing, keep exploring, and most importantly, keep having fun with math!