Arithmetic Series: Sum Of First 25 Terms

Let's tackle this arithmetic series problem step by step! Guys, if you're scratching your head over sequences and series, don't worry; we'll break it down so it's super easy to understand. We're given that the 4th term (${a_4}$) of an arithmetic sequence is 32 and the 8th term (${a_8}$) is 48. Our mission is to find the sum of the first 25 terms (${S_{25}}$) of this sequence. Sounds like fun, right?

Understanding Arithmetic Sequences

First, let's recap what an arithmetic sequence actually is. An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference, usually denoted by ${d}$. The ${n}$-th term of an arithmetic sequence can be expressed as:

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

where:

• ${a_n}$ is the ${n}$-th term,
• ${a_1}$ is the first term,
• ${n}$ is the term number, and
• ${d}$ is the common difference.

Finding the First Term and Common Difference

Okay, back to our problem. We know ${a_4 = 32}$ and ${a_8 = 48}$. Using the formula for the ${n}$-th term, we can write two equations:

1. ${a_4 = a_1 + 3d = 32}$
2. ${a_8 = a_1 + 7d = 48}$

Now we have a system of two linear equations with two variables, ${a_1}$ and ${d}$. We can solve this system using substitution or elimination. Let's use elimination. Subtract the first equation from the second equation:

${(a_1 + 7d) - (a_1 + 3d) = 48 - 32}$

${4d = 16}$

${d = 4}$

Great! We found the common difference, which is 4. Now, we can substitute the value of ${d}$ back into one of the equations to find ${a_1}$. Let's use the first equation:

${a_1 + 3(4) = 32}$

${a_1 + 12 = 32}$

${a_1 = 32 - 12}$

${a_1 = 20}$

So, the first term, ${a_1}$, is 20. Now we know both the first term and the common difference. Awesome!

Calculating the Sum of the First 25 Terms

Next, we need to find the sum of the first 25 terms of the arithmetic series. The formula for the sum of the first ${n}$ terms of an arithmetic series is:

${S_n = \frac{n}{2}(2a_1 + (n - 1)d)}$

In our case, we want to find ${S_{25}}$, so ${n = 25}$, ${a_1 = 20}$, and ${d = 4}$. Plugging these values into the formula, we get:

${S_{25} = \frac{25}{2}(2(20) + (25 - 1)(4))}$

${S_{25} = \frac{25}{2}(40 + 24(4))}$

${S_{25} = \frac{25}{2}(40 + 96)}$

${S_{25} = \frac{25}{2}(136)}$

${S_{25} = 25 \times 68}$

${S_{25} = 1700}$

Final Answer

Therefore, the sum of the first 25 terms of the arithmetic series is 1700. So the correct option is C. 1700. High five! We nailed it!

Key points to remember:

• The formula for the ${n}$-th term of an arithmetic sequence: ${a_n = a_1 + (n - 1)d}$
• The formula for the sum of the first ${n}$ terms of an arithmetic series: ${S_n = \frac{n}{2}(2a_1 + (n - 1)d)}$

Why This Matters

You might be wondering, why even bother with arithmetic series? Well, arithmetic sequences and series pop up in various real-world scenarios. For instance, calculating simple interest, predicting evenly increasing costs, or even in some basic physics problems. Understanding these concepts gives you a solid foundation for more advanced math and problem-solving.

Let's dive deeper into why understanding arithmetic sequences and series is super useful and how they relate to more advanced math. When you grasp the core principles of arithmetic sequences, you are essentially building a fundamental block for tackling more complex mathematical concepts. These sequences aren't just abstract numbers; they represent patterns that appear everywhere.

Building Blocks for Advanced Math

Arithmetic sequences lay the groundwork for understanding series, which in turn are essential for calculus. In calculus, you'll encounter infinite series, which are used to approximate functions, solve differential equations, and model various physical phenomena. Without a solid understanding of basic arithmetic series, grasping these advanced concepts becomes significantly harder. Think of it as trying to build a skyscraper without a strong foundation—it just won't work.

Real-World Applications

Beyond theoretical math, arithmetic sequences and series have numerous practical applications:

• Finance: Simple interest calculations are a direct application of arithmetic sequences. Understanding how a fixed amount of interest accumulates over time is crucial for making informed financial decisions.
• Physics: Uniformly accelerated motion can be modeled using arithmetic sequences. For example, the distance traveled by an object with constant acceleration increases in an arithmetic progression.
• Computer Science: In programming, loops often involve arithmetic sequences. For instance, iterating through an array with a fixed step size creates an arithmetic sequence of indices.

Problem-Solving Skills

Working with arithmetic sequences and series sharpens your problem-solving skills. You learn to identify patterns, formulate equations, and solve systems of equations. These skills are transferable to many other areas of life, from engineering to economics.

Examples in Action

Consider a simple example: Suppose you save $100 each month. The total amount you save each month forms an arithmetic sequence: $100, $200, $300, and so on. If you want to know how much you'll have saved after 12 months, you can use the formula for the sum of an arithmetic series to quickly calculate the total.

Tying it All Together

So, next time you encounter an arithmetic sequence problem, remember that you're not just crunching numbers. You're reinforcing foundational math skills, learning to recognize patterns, and building a toolkit for solving real-world problems. Keep practicing, and you'll find that these concepts become second nature!

And remember, mathematics is not just about formulas; it's about understanding the underlying logic and applying it to solve problems efficiently. So, keep practicing and exploring, and you'll become a math whiz in no time! Keep your eye on the pattern and you will solve the problem.