Finding The 25th Term Of An Arithmetic Series

Hey guys! Let's dive into a classic math problem involving arithmetic series. We're given some cool information: the sum of the first 20 terms is 860, and the 5th term is 32. Our mission? To find the 25th term of this series. Sounds like fun, right? This is a great example of how we can use the properties of arithmetic sequences and series to solve for specific terms. Arithmetic series problems are fundamental in mathematics, and understanding them is super important for anyone studying algebra or precalculus. We'll break this down step by step, making sure everyone understands the process. Get ready to flex those math muscles! We will review the formulas and some basic concepts to solve this problem.

First, let's refresh our memories on what an arithmetic series actually is. In simple terms, it's a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. Each term in the series is obtained by adding the common difference to the previous term. The sum of an arithmetic series is the sum of all the terms in the series. The sum of the first 'n' terms of an arithmetic series, usually denoted by 'Sn', can be calculated using a few different formulas, which we'll explore. It's like a staircase where each step is the same height apart; each term is equally distanced from the ones around it. The main idea behind solving arithmetic series problems like this is to carefully identify the given information and choose the right formulas to find the unknown values. The formulas help us connect the knowns (like the sum of the first 20 terms or a specific term) with the unknowns (like the common difference or a specific term we want to find).

To solve this particular problem, we're going to need a couple of key formulas in our arsenal. We will use the formula for the sum of an arithmetic series, and also the formula for the nth term. The formula for the sum of the first 'n' terms (Sn) of an arithmetic series is: Sn = n/2 * [2a + (n-1)d], where 'n' is the number of terms, 'a' is the first term, and 'd' is the common difference. And the formula to find the nth term (an) of an arithmetic series is: an = a + (n-1)d. With these tools, we're set to tackle the problem systematically. The formulas are our trusty sidekicks. We will carefully apply them to find the unknown terms. Remember, the key is to isolate what we need to find, use the correct formula, and make sure we perform the arithmetic operations carefully. Before we get into the calculations, always take a moment to understand what the problem is asking. Identify the known values and what you need to figure out. This approach will make the whole process much easier and reduce errors. Now, let’s go through the problem step by step to find the 25th term.

Step-by-Step Solution

Alright, let's break this down bit by bit. We've got two main pieces of information that we can use to begin. First, we know that the sum of the first 20 terms (S20) is 860. Second, we know that the 5th term (a5) is 32. We can use these facts to find the common difference and the first term, which will then allow us to find the 25th term. This will involve using the formulas mentioned earlier and solving a system of equations.

Using the Sum Formula

Let’s start with the sum of the first 20 terms (S20 = 860). Using the formula Sn = n/2 * [2a + (n-1)d], and plugging in n = 20, we get:

860 = 20/2 * [2a + (20-1)d]

This simplifies to:

860 = 10 * [2a + 19d]

Further simplifying gives us:

86 = 2a + 19d (Equation 1)

This is our first equation, which we can use later. We've used the sum formula to establish a relationship between the first term (a) and the common difference (d). This is a vital first step, because it gives us an equation that we can use to solve for our unknowns. We now have a relationship. Remember, the more information you can get into a usable form, the easier it is to finally solve a problem of this kind. So let’s not stop there, and move onto the next piece of info.

Using the nth Term Formula

Now, let's look at the 5th term (a5 = 32). Using the formula an = a + (n-1)d, and plugging in n = 5, we get:

32 = a + (5-1)d

This simplifies to:

32 = a + 4d (Equation 2)

This gives us our second equation. It creates another relationship between 'a' and 'd'. Now that we have two equations, we can solve them simultaneously to find the values of 'a' (the first term) and 'd' (the common difference). This is where our algebra skills come into play. We are able to do this through the combination of the two equations.

Solving the System of Equations

We have two equations:

Equation 1: 86 = 2a + 19d Equation 2: 32 = a + 4d

We can solve this system of equations. Let's solve Equation 2 for 'a':

a = 32 - 4d

Now, substitute this value of 'a' into Equation 1:

86 = 2(32 - 4d) + 19d

Expanding gives us:

86 = 64 - 8d + 19d

Combining like terms:

86 = 64 + 11d

Subtracting 64 from both sides:

22 = 11d

Therefore:

d = 2

Now that we have 'd', we can find 'a'. Substitute d = 2 into the equation a = 32 - 4d:

a = 32 - 4(2)

a = 32 - 8

a = 24

So, the first term (a) is 24, and the common difference (d) is 2. Excellent! We've found the building blocks we need to proceed. Solving the system of equations is a critical step, because it gives us the values of both the first term and the common difference. We can move on to the final step where we will use these numbers to get to the solution we're looking for.

Finding the 25th Term

Now that we know 'a' (the first term) is 24 and 'd' (the common difference) is 2, we can find the 25th term (a25) using the formula an = a + (n-1)d. Plugging in n = 25:

a25 = 24 + (25-1) * 2

a25 = 24 + 24 * 2

a25 = 24 + 48

a25 = 72

Therefore, the 25th term of the arithmetic series is 72. We did it! We have successfully determined the 25th term. This is the goal of our mission. It’s important to note the power of understanding the basic formulas that let us solve the problem systematically. Each step is critical and helps us to understand how we can approach other problems like this. We combined the knowledge of the sum formula and the nth term formula to solve the problem.

Conclusion

So, there you have it, guys! We started with some information about an arithmetic series and managed to find the 25th term using a step-by-step approach. By understanding the formulas and the relationship between the terms, we could solve this problem efficiently. Remember, practice is key! The more you work through problems like these, the better you'll become at recognizing the patterns and applying the formulas. Keep up the great work, and happy math-ing! Arithmetic series problems might seem tricky at first, but with practice, they become quite manageable. The process we used here can be adapted to other similar problems, so make sure you understand each step. Mastering these skills builds a strong foundation for future mathematical endeavors. Remember, the key is to stay organized, apply the right formulas, and take it one step at a time.