Arithmetic Sequence: Find S(10) - S(8) If S(3) = 3 & S(6) = 24

Nov 19, 2025 by Andrew McMorgan 63 views

Hey Plastik Magazine readers! Today, we're diving deep into the fascinating world of arithmetic sequences. If you've ever geeked out over patterns in numbers, or simply enjoy a good mathematical brain-teaser, you're in for a treat. We're going to break down a problem that looks tricky at first glance, but trust me, with a little bit of algebraic finesse, it's totally solvable. Our mission? To find the difference between the 10th and 8th terms ( $S_{10} - S_8$ ) in an arithmetic sequence, given that the sum of the first three terms ( $S_3$ ) is 3 and the sum of the first six terms ( $S_6$ ) is 24. Sounds like fun, right? Let's jump right in and unravel this mathematical puzzle together! We will guide you step by step to understand arithmetic sequences and solve this problem effectively.

Understanding Arithmetic Sequences

First, let's rewind a bit and make sure we're all on the same page about what an arithmetic sequence actually is. An arithmetic sequence, at its heart, is just a list of numbers where the difference between any two consecutive terms is constant. Think of it like climbing stairs – each step you take is the same height, so you're consistently adding that height to your elevation. This constant difference is what mathematicians call the 'common difference,' often denoted by 'd'. So, if you start with a number (let's call it 'a', the first term) and keep adding 'd', you'll generate the entire sequence. For example, if a = 2 and d = 3, your sequence would be 2, 5, 8, 11, and so on. Each term is simply the previous term plus 3. Now, here's where it gets interesting. We often want to talk about the sum of the first 'n' terms of such a sequence. This sum is usually written as $S_n$ . There's a neat little formula for calculating $S_n$ directly, which saves us from having to add up all the terms individually. This formula connects the sum to the first term ('a'), the number of terms ('n'), and the common difference ('d'). Understanding this relationship is crucial for tackling problems like the one we're about to solve. So, remember, arithmetic sequences are all about consistent steps, and their sums have a predictable pattern. The formula for the sum of an arithmetic sequence is a powerful tool in our mathematical arsenal. We'll be using it shortly to crack our problem wide open. Knowing the properties of arithmetic sequences such as the common difference and how each term relates to each other is crucial for solving this type of problem. Let’s keep these basics in mind as we proceed. The following sections will detail the application of these concepts in solving the given problem.

Setting Up the Equations

Okay, now that we've refreshed our understanding of arithmetic sequences, let's get our hands dirty with the problem at hand. We're given two key pieces of information: $S_3 = 3$ and $S_6 = 24$ . Remember, $S_n$ represents the sum of the first 'n' terms of our sequence. So, $S_3$ is the sum of the first three terms, and $S_6$ is the sum of the first six terms. To really make use of these facts, we need to express them in terms of our arithmetic sequence formula. As we discussed earlier, the sum of the first 'n' terms of an arithmetic sequence can be calculated using the formula: $S_n = \frac{n}{2}[2a + (n-1)d]$ , where 'a' is the first term and 'd' is the common difference. Let’s apply this formula to our given information. For $S_3 = 3$ , we can write: $3 = \frac{3}{2}[2a + 2d]$ . Notice how we've substituted n = 3 into the formula. This equation relates 'a' and 'd', giving us our first crucial link. Similarly, for $S_6 = 24$ , we get: $24 = \frac{6}{2}[2a + 5d]$ . Here, we've plugged in n = 6, creating another equation connecting 'a' and 'd'. Now we have a system of two equations with two unknowns ('a' and 'd'). This is fantastic news because it means we can solve for these variables using techniques from algebra. These two equations are the backbone of our solution. By solving them simultaneously, we'll unlock the values of 'a' and 'd', which are the building blocks of our arithmetic sequence. Once we know 'a' and 'd', we can find any term in the sequence, and, importantly, calculate $S_{10} - S_8$ . So, stay tuned, because the next step is where we actually solve this system and reveal the values of 'a' and 'd'. Remember, setting up the equations correctly is half the battle in any math problem, and we've just nailed that part! The proper setup allows us to leverage algebraic techniques effectively.

Solving for 'a' and 'd'

Alright, guys, we've set up our equations, and now it's time for some algebraic magic! We have two equations:

$3 = \frac{3}{2}[2a + 2d]$
$24 = \frac{6}{2}[2a + 5d]$

Let's simplify these a bit to make them easier to work with. We can divide both sides of the first equation by 3 to get:

$1 = \frac{1}{2}[2a + 2d]$

Multiplying both sides by 2, we get:

$2 = 2a + 2d$

And dividing by 2 again, we arrive at our simplified first equation:

$1 = a + d$ (Equation 1 Simplified)

Now, let's tackle the second equation. First, simplify the fraction:

$24 = 3[2a + 5d]$

Divide both sides by 3:

$8 = 2a + 5d$ (Equation 2 Simplified)

Now we have a much cleaner system of equations:

$1 = a + d$
$8 = 2a + 5d$

There are a couple of ways we could solve this system – substitution or elimination. Let's use the substitution method. From Equation 1 Simplified, we can easily express 'a' in terms of 'd':

$a = 1 - d$

Now, substitute this expression for 'a' into Equation 2 Simplified:

$8 = 2(1 - d) + 5d$

Expand and simplify:

$8 = 2 - 2d + 5d$

$8 = 2 + 3d$

Subtract 2 from both sides:

$6 = 3d$

Divide by 3:

$d = 2$

Woohoo! We've found the common difference, d = 2. Now, let's plug this value back into our expression for 'a':

$a = 1 - d = 1 - 2$

$a = -1$

So, we've cracked it! The first term, a, is -1, and the common difference, d, is 2. This is a major victory, because now we know the complete blueprint for our arithmetic sequence. Knowing the values of 'a' and 'd' allows us to calculate any term or sum of terms in the sequence. This is a crucial step towards finding the final answer. We've successfully navigated the algebraic maze and emerged with the key pieces of information needed to solve the problem. The next step is to use these values to find $S_{10}$ and $S_8$ , and ultimately, their difference.

Calculating S(10) and S(8)

Okay, team, we've got 'a' and 'd' locked down. We know the first term is -1 and the common difference is 2. Now comes the fun part – calculating $S_{10}$ and $S_8$ . Remember, these represent the sum of the first 10 terms and the sum of the first 8 terms, respectively. To find these, we'll dust off our arithmetic series sum formula once more: $S_n = \frac{n}{2}[2a + (n-1)d]$ . Let's start with $S_{10}$ . We'll plug in n = 10, a = -1, and d = 2:

$S_{10} = \frac{10}{2}[2(-1) + (10-1)2]$

Simplify:

$S_{10} = 5[-2 + 18]$

$S_{10} = 5[16]$

$S_{10} = 80$

Awesome! The sum of the first 10 terms is 80. Now, let's do the same for $S_8$ . This time, we'll plug in n = 8, a = -1, and d = 2:

$S_8 = \frac{8}{2}[2(-1) + (8-1)2]$

Simplify:

$S_8 = 4[-2 + 14]$

$S_8 = 4[12]$

$S_8 = 48$

Fantastic! The sum of the first 8 terms is 48. We're in the home stretch now. We've calculated $S_{10}$ and $S_8$ individually. The problem asks us for the difference between these two sums. So, all that's left to do is subtract! Remember, finding the values of $S_{10}$ and $S_8$ required careful application of the arithmetic series sum formula, ensuring we correctly substituted the values of 'n', 'a', and 'd'. This meticulous approach is key to avoiding errors and arriving at the correct answer. Now that we have these values, the final calculation is straightforward. Let’s proceed to the last step and wrap up this problem.

Finding S(10) - S(8)

Okay, guys, the moment we've been working towards is finally here! We've successfully calculated $S_{10}$ and $S_8$ . We know that $S_{10} = 80$ and $S_8 = 48$ . The original question asked us to find the value of $S_{10} - S_8$ . This is a simple subtraction problem now:

$S_{10} - S_8 = 80 - 48$

$S_{10} - S_8 = 32$

And there you have it! The value of $S_{10} - S_8$ is 32. We've successfully navigated through the entire problem, from understanding arithmetic sequences to setting up equations, solving for 'a' and 'd', calculating the sums, and finally, finding the difference. This problem beautifully illustrates how different concepts in math come together to solve a single question. We used the definition of an arithmetic sequence, the formula for the sum of an arithmetic series, and some good old-fashioned algebra to reach our solution. It’s a testament to the power of a systematic approach and a solid understanding of fundamental principles. So, what does this answer mean in the context of the arithmetic sequence? Remember that $S_{10} - S_8$ represents the sum of just the 9th and 10th terms of the sequence. We've effectively found that the 9th term plus the 10th term equals 32. This final step underscores the importance of understanding what the mathematical notation represents in real terms. We've not just crunched numbers; we've gained insight into the behavior of this specific arithmetic sequence. Congratulations, everyone! We’ve tackled this problem head-on and emerged victorious. Remember, the key to success in math is breaking down complex problems into smaller, manageable steps. And that’s exactly what we did here.