Arithmetic Sequence: Find The Common Difference

Nov 12, 2025 by ADMIN 48 views

Hey guys! Today, we're diving into a classic math problem involving arithmetic sequences. We're going to break down how to find the common difference when you're given the first term and another term further down the sequence. It might sound tricky, but trust me, it's totally manageable. So, let's get started and unravel this arithmetic mystery together!

Understanding Arithmetic Sequences

Before we jump into solving the problem, let's make sure we're all on the same page about what an arithmetic sequence actually is. An arithmetic sequence is simply a list of numbers where the difference between any two 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 stairs where each step is the same height. The height you gain with each step is the common difference. For example, the sequence 2, 4, 6, 8, 10... is an arithmetic sequence because we're adding 2 each time (the common difference is 2). Similarly, 10, 7, 4, 1, -2... is also an arithmetic sequence, but here we're subtracting 3 each time (the common difference is -3).

Knowing this, let's put it into formula form. The nth term of an arithmetic sequence ( $a_n$ ) can be expressed using the first term ( $a_1$ ) and the common difference (d) as follows:

$a_n = a_1 + (n - 1)d$

This formula is super important because it allows us to find any term in the sequence if we know the first term and the common difference. We'll be using this formula to solve our problem today, so make sure you've got it locked in!

Problem Setup

Okay, let's get down to the specific problem we're tackling. We're given an arithmetic sequence where the first term, denoted as $a_1$ , is equal to 2. We also know that the sixth term, denoted as $a_6$ , is equal to -1/2. Our mission, should we choose to accept it (and we do!), is to find the common difference, 'd', of this sequence.

So, to recap, we have:

$a_1 = 2$
$a_6 = - rac{1}{2}$
We need to find 'd'

Now that we have all the pieces of the puzzle laid out, we can start thinking about how to put them together to find our solution. Remember that formula we talked about earlier? That's going to be our key to unlocking this problem. We know the value of $a_6$ , and we know its position in the sequence (n = 6). We also know $a_1$ . So, we have everything we need to plug into the formula and solve for 'd'. Let's do it!

Applying the Formula

Remember our formula for the nth term of an arithmetic sequence? It's:

$a_n = a_1 + (n - 1)d$

In our case, we know $a_6$ , so we can plug in n = 6 into the formula:

$a_6 = a_1 + (6 - 1)d$

Now, let's substitute the values we know. We have $a_1 = 2$ and $a_6 = - rac{1}{2}$ . Plugging these in, we get:

$- rac{1}{2} = 2 + (5)d$

See how we've transformed the problem? We've gone from a sequence question to a simple algebraic equation. Now, all we need to do is solve for 'd'. This is where our algebra skills come into play. We'll isolate 'd' on one side of the equation to find its value. Are you ready to take the next step? Let's dive into the algebra!

Solving for the Common Difference

Alright, let's solve the equation we got in the last step:

$- rac{1}{2} = 2 + 5d$

Our goal is to isolate 'd'. The first step is to get rid of the 2 on the right side of the equation. We can do this by subtracting 2 from both sides:

$- rac{1}{2} - 2 = 5d$

Now, we need to simplify the left side. To subtract 2 from -1/2, we need to express 2 as a fraction with a denominator of 2. So, 2 becomes 4/2:

$- rac{1}{2} - rac{4}{2} = 5d$

Combining the fractions, we get:

$- rac{5}{2} = 5d$

Now, we're almost there! To isolate 'd', we need to divide both sides of the equation by 5. Remember that dividing by 5 is the same as multiplying by 1/5:

$- rac{5}{2} * rac{1}{5} = d$

Multiplying the fractions, we get:

$- rac{5}{10} = d$

Finally, we can simplify the fraction by dividing both the numerator and denominator by 5:

$d = - rac{1}{2}$

And there we have it! We've successfully found the common difference. Feels good, right? Let's recap our journey and make sure we fully understand the solution.

Conclusion

So, after all that work, we've found that the common difference, 'd', of the arithmetic sequence is -1/2. That means that to get from one term to the next in this sequence, you subtract 1/2. Cool, huh?

Let's quickly recap the steps we took to solve this problem:

We understood the definition of an arithmetic sequence and the concept of a common difference.
We remembered the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n - 1)d$ .
We plugged in the given values ( $a_1 = 2$ , $a_6 = - rac{1}{2}$ , and n = 6) into the formula.
We solved the resulting algebraic equation for 'd'.

This type of problem is a great example of how math concepts build upon each other. You need to understand the basic definition of an arithmetic sequence, know the formula, and then be able to apply your algebra skills to solve for the unknown.

I hope this explanation was clear and helpful! Arithmetic sequences are a fundamental topic in math, and mastering them will open doors to more advanced concepts down the road. Keep practicing, and you'll become an arithmetic sequence pro in no time!