Solving A Sequence Problem: Find 2b + A

Hey guys! Today, we're diving into a cool math problem involving sequences. Sequences can seem tricky at first, but once you understand the underlying concepts, they become super manageable. We're going to break down a problem step-by-step, so you can tackle similar questions with confidence. This article is aimed to provide a comprehensive guide on how to solve sequence problems, focusing on a specific example but also providing general strategies that can be applied to various scenarios. So, buckle up and let's get started!

Understanding the Problem

The problem states: In the sequence a < b < c < 15, the differences between consecutive numbers are equal. We need to find the value of the expression 2 * b + a.

This means we have an arithmetic sequence where the numbers a, b, and c are equally spaced. The key here is that the difference between a and b is the same as the difference between b and c. Understanding this equal spacing is crucial for solving the problem. We also know that all these numbers are less than 15, which gives us a boundary to work within. Let's break down each component to really grasp what's happening.

• Arithmetic Sequence: This is our main concept. An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. This constant difference is often called the 'common difference.' Think of it like climbing stairs; each step is the same height.
• Consecutive Numbers: These are numbers that follow each other in order, without any gaps. For example, 1, 2, and 3 are consecutive numbers. In our sequence, a, b, and c are consecutive in the sense that they follow each other with a constant difference.
• Equal Differences: This is the heart of the problem. The difference between a and b is the same as the difference between b and c. Mathematically, this can be written as b - a = c - b. This equality is what allows us to set up equations and solve for the unknowns.
• Expression 2 * b + a: This is what we ultimately need to find. Once we know the values of a and b, we can simply plug them into this expression to get our answer. It’s like the final destination of our mathematical journey!

By clearly understanding these components, we can formulate a solid plan to attack the problem. The next step is to translate this understanding into mathematical terms and start solving for our unknowns. Keep reading, and we'll walk through the solution together!

Setting Up the Equations

To solve this sequence problem effectively, let's translate the given information into mathematical equations. This is where the real problem-solving magic begins! By expressing the relationships between the numbers as equations, we can use algebraic techniques to find the values of a, b, and c. So, let’s dive into how we can set up these equations.

Since the differences between consecutive numbers are equal, we can express this relationship mathematically. Let's denote the common difference between the numbers as d. This means that the difference between a and b is d, and the difference between b and c is also d. We can write these relationships as follows:

• b = a + d
• c = b + d

These two equations capture the essence of the problem: the constant difference between consecutive terms. The first equation, b = a + d, tells us that b is simply a plus the common difference d. Similarly, the second equation, c = b + d, tells us that c is b plus the same common difference d. These equations are the foundation upon which we'll build our solution.

But we can take it a step further and express c in terms of a and d. Since c = b + d and b = a + d, we can substitute the expression for b into the equation for c. This gives us:

c = (a + d) + d c = a + 2d

Now we have c expressed in terms of a and d, which will be incredibly useful in the next steps. We now have a system of equations that relates a, b, c, and d. These equations allow us to manipulate the variables and eventually solve for the values we need.

Why is this important? By setting up these equations, we’ve transformed a word problem into a concrete mathematical problem. Instead of just thinking about the relationships, we can now use the power of algebra to manipulate these relationships and find solutions. It’s like having a map that guides us through the problem-solving process.

The next step is to use the additional information provided in the problem – that c is less than 15 – and explore integer values to find possible solutions for a and d. This is where we start narrowing down the possibilities and getting closer to the answer. So keep reading to see how we put these equations to work!

Finding Possible Values

Now that we have our equations set up, the next step is to find possible values for a, b, c, and d. Remember, the problem states that a < b < c < 15, and the differences between the numbers are equal. This gives us a crucial constraint: all our numbers must be less than 15. This constraint is like a fence that keeps our solutions within a certain range, making our task much easier. We're essentially looking for integer solutions that fit our criteria.

We know that c = a + 2d, and c < 15. This is a key piece of information. Since a and d must be positive integers (because a < b < c), we can start testing values for d and see what values of a would result in c being less than 15.

Let's start by trying different values for d, the common difference, and see what values of a we can find that satisfy the condition c < 15. This is where we put on our detective hats and start exploring the possibilities!

• If d = 1, then c = a + 2. Since c < 15, we have a + 2 < 15, which means a < 13. Possible values for a could be 1, 2, 3, ..., 12.
• If d = 2, then c = a + 4. Since c < 15, we have a + 4 < 15, which means a < 11. Possible values for a could be 1, 2, 3, ..., 10.
• If d = 3, then c = a + 6. Since c < 15, we have a + 6 < 15, which means a < 9. Possible values for a could be 1, 2, 3, ..., 8.
• If d = 4, then c = a + 8. Since c < 15, we have a + 8 < 15, which means a < 7. Possible values for a could be 1, 2, 3, ..., 6.
• If d = 5, then c = a + 10. Since c < 15, we have a + 10 < 15, which means a < 5. Possible values for a could be 1, 2, 3, 4.
• If d = 6, then c = a + 12. Since c < 15, we have a + 12 < 15, which means a < 3. Possible values for a could be 1, 2.
• If d = 7, then c = a + 14. Since c < 15, we have a + 14 < 15, which means a < 1. But a must be a positive integer, so there are no possible values for a in this case.

So, we've systematically explored different possibilities for d and found the corresponding range of values for a. This is a powerful technique in problem-solving: breaking down the problem into smaller, manageable chunks and exploring each one.

Now, we need to consider another critical factor: the numbers a, b, and c must be distinct (different from each other). This condition will help us narrow down the possibilities even further. We're not just looking for any values of a and d; we need values that will give us distinct a, b, and c that fit within our sequence. The next step will be focusing on a specific value of d and testing different values of a to see if they meet all our conditions. So stick around, we're getting closer to the solution!

Testing Values and Finding the Solution

Alright, guys, we've done a fantastic job narrowing down the possibilities! We've got our equations, we've explored the ranges for a and d, and now it's time to put on our detective hats again and test some values. This is where the puzzle pieces start to come together, and we get closer to finding our solution. We need to test the values we found for a and d to ensure they produce distinct values for a, b, and c, all less than 15.

Let's pick a value for d and systematically test values for a. A good starting point might be d = 4 because it gives us a manageable range of possible a values (1 to 6). Remember, when d = 4, we have c = a + 8.

• If a = 1, then b = a + d = 1 + 4 = 5, and c = a + 2d = 1 + 8 = 9. This gives us the sequence 1 < 5 < 9 < 15. This looks promising!
• If a = 2, then b = a + d = 2 + 4 = 6, and c = a + 2d = 2 + 8 = 10. This gives us the sequence 2 < 6 < 10 < 15. Another good candidate!
• If a = 3, then b = a + d = 3 + 4 = 7, and c = a + 2d = 3 + 8 = 11. This gives us the sequence 3 < 7 < 11 < 15. Still looking good!
• If a = 4, then b = a + d = 4 + 4 = 8, and c = a + 2d = 4 + 8 = 12. This gives us the sequence 4 < 8 < 12 < 15. This works too!
• If a = 5, then b = a + d = 5 + 4 = 9, and c = a + 2d = 5 + 8 = 13. This gives us the sequence 5 < 9 < 13 < 15. Yes!
• If a = 6, then b = a + d = 6 + 4 = 10, and c = a + 2d = 6 + 8 = 14. This gives us the sequence 6 < 10 < 14 < 15. Fantastic!

So, when d = 4, we found several possible sequences. Now, let's take the first sequence we found (a = 1, b = 5, c = 9) and calculate the value of the expression 2 * b + a.

2 * b + a = 2 * 5 + 1 = 10 + 1 = 11

Okay, we have a potential answer: 11. But before we declare victory, we should check if this value holds true for other valid sequences. This is crucial because we want to ensure our solution is consistent and not just a lucky guess.

Let’s test the next sequence (a = 2, b = 6, c = 10):

2 * b + a = 2 * 6 + 2 = 12 + 2 = 14

Uh oh! We got a different answer. This means our initial assumption that any valid sequence will give us the same result for 2 * b + a was incorrect. This is a valuable lesson in problem-solving: always verify your assumptions! This difference indicates there may be a constraint or condition we haven't fully considered or that the problem may have multiple valid solutions for a, b, and c but only one value for the expression 2b + a. We need to go back and examine the problem statement more closely.

Looking back at the problem, we realize there may be a typo! It's highly likely that the problem intended to specify a unique solution, and the fact that we're getting different values suggests an issue with the problem statement itself. Let's assume, for the sake of continuing the problem-solving process, that the first valid sequence we found is the intended solution. This is a common situation in real-world problem-solving: sometimes, you need to make educated guesses and state your assumptions clearly.

So, based on the sequence a = 1, b = 5, c = 9, we found that 2 * b + a = 11.

Final Answer and Conclusion

Alright, guys, after a thorough investigation and some careful calculations, we've arrived at our final answer. Remember, we started with a sequence problem where the differences between consecutive numbers were equal, and we needed to find the value of the expression 2 * b + a. We set up equations, explored possible values, tested those values, and then calculated our final result.

Based on our initial assumption and calculations using the sequence a = 1, b = 5, and c = 9, we found that:

2 * b + a = 2 * 5 + 1 = 11

So, our final answer is 11.

But remember, during our problem-solving journey, we encountered a discrepancy. We found that different valid sequences resulted in different values for the expression 2 * b + a. This led us to suspect a possible typo or ambiguity in the problem statement. It’s a crucial lesson in mathematics and problem-solving: always be critical, check your assumptions, and be prepared to question the problem itself if something doesn't seem right.

Key Takeaways:

• Break down the problem: We started by understanding the core concepts and breaking the problem into smaller, manageable parts. This made the problem less intimidating and easier to tackle.
• Translate to equations: We transformed the word problem into mathematical equations. This allowed us to use algebraic techniques to solve for the unknowns.
• Explore and test: We systematically explored possible values for the variables and tested them against the given conditions. This helped us narrow down the solutions and find the correct one.
• Verify and question: We verified our solution by testing it with different sequences. This led us to uncover a potential issue with the problem statement, highlighting the importance of critical thinking and questioning assumptions.

This problem wasn't just about finding the right answer; it was about the process of problem-solving. We learned how to approach a problem systematically, how to translate words into math, and how to think critically about our solutions. These skills are valuable not just in math, but in all areas of life.

So, guys, keep practicing, keep questioning, and keep solving! You've got this!