Arithmetic Progression: Finding The Common Difference & First Term
Hey guys! Let's dive into a classic math problem involving arithmetic progressions (APs). We've got a scenario where we know a couple of terms in the sequence, and our mission is to figure out the common difference (the 'reason' in the original prompt) and the very first term. It's like a mathematical puzzle, and I'm here to walk you through it step by step. So, grab your pencils, and let's get started!
Understanding Arithmetic Progressions
First things first: what exactly is an arithmetic progression? Well, it's simply a sequence of numbers where the difference between consecutive terms is constant. This constant difference is what we call the common difference, often denoted by the letter 'd'. Think of it like climbing a staircase where each step is the same height; you're adding the same amount each time. For example, the sequence 2, 5, 8, 11... is an AP, with a common difference of 3. Each term is obtained by adding 3 to the previous term. The general form of an arithmetic progression is: a, a + d, a + 2d, a + 3d, ... where 'a' is the first term.
So, if we know two terms in an AP, we can actually deduce all sorts of information about the sequence. Knowing a4 and a9 lets us find the common difference easily. The formula for the nth term of an AP is an = a + (n-1)d. This formula is super important and we will use it for solving the problem. The a in the formula represents the first term, n is the number of the term we're looking for, and d, of course, is the common difference. The beauty of this formula is that it gives us a direct way to calculate any term if we know the first term and the common difference. And, as we'll see, even if we don't know a and d initially, we can still often figure them out using the information we do have.
Now, let's look back at our problem. We know a4 is 12 and a9 is 27. This means the fourth term in the sequence is 12, and the ninth term is 27. Using this information, we can solve for d and a. Ready? Let's go!
Solving for the Common Difference (d)
Alright, let's roll up our sleeves and tackle finding that common difference, d. This is often the first step in solving these types of problems. Remember, the common difference is the constant amount that we add (or subtract) to get from one term to the next in the sequence. We know a4 = 12 and a9 = 27. This information is our key to unlocking the solution!
Think about it this way: to get from the 4th term to the 9th term, we add the common difference a certain number of times. Specifically, we add the common difference 5 times (9 - 4 = 5). So, the difference between a9 and a4 is equal to 5 times the common difference (5*d). We can write this relationship as:
a9 - a4 = 5d
Now, let's plug in the values we know:
27 - 12 = 5d
Simplifying this, we get:
15 = 5d
To isolate d, we divide both sides of the equation by 5:
d = 15 / 5
Therefore:
d = 3
So, we've found our common difference: d = 3. This means that to get from one term to the next in this arithmetic progression, we add 3. Awesome, right? We're halfway there! Knowing the common difference is a huge step toward understanding the entire sequence.
Now that we know the common difference is 3, we can confidently move on to find the first term, which is the starting point of our sequence. Ready to do it? Let's go!
Finding the First Term (a)
Now that we've successfully calculated the common difference (d = 3), it's time to find the first term, which we'll denote as 'a'. Remember, the first term is the starting point of the sequence. We have a few ways we can do this, but the easiest is using the formula for the nth term: an = a + (n-1)d. Since we know d and have two terms, we can pick either one to calculate the first term. Let's use a4 = 12.
We know a4 = 12, n = 4, and d = 3. Substitute these values into the formula:
12 = a + (4 - 1) * 3
Simplify the equation:
12 = a + (3) * 3
12 = a + 9
To isolate 'a', subtract 9 from both sides:
12 - 9 = a
Therefore:
a = 3
So, the first term, a, is 3. That means our arithmetic progression starts with the number 3. The sequence would be: 3, 6, 9, 12... and so on. We did it! We have successfully determined both the common difference and the first term of the arithmetic progression. Great job, guys! This kind of problem is fundamental to understanding sequences and series in mathematics.
Conclusion: Putting It All Together
Alright, let's recap what we've accomplished. We started with the knowledge that a4 = 12 and a9 = 27 in an arithmetic progression. From this, we successfully determined:
- The common difference (d) = 3.
- The first term (a) = 3.
We used the relationship between the terms and the concept of a constant difference to solve for d. Then, we used the formula for the nth term of an arithmetic progression to find a. It's all about using the known information and applying the correct formulas and relationships.
This kind of problem is a cornerstone in understanding sequences and series. Being able to find the common difference and the first term allows us to fully describe the entire progression and predict future terms. It's a fundamental concept that you'll encounter often in your math journey. Keep practicing, and you'll become a pro at these problems! If you got any questions, feel free to ask. Keep up the great work!
Answer
Based on our calculations:
- Reason (Common Difference): 3
- First term: 3
Looking at the options provided, the correct answer should be: The prompt appears to be incorrect as the values do not match any of the provided options. The correct values are: Reason: 3, First term: 3. Hence, none of the options given (a, b, c, or d) match the correct answer.