Arithmetic Series: Sum Of 12 Terms Explained

Hey guys! Let's dive into a cool math problem involving arithmetic series. We've got a sequence with 12 terms, and some clues about the sums of the first few and last few terms. Our mission? To find the sum of all the terms in the series. Sounds like a fun challenge, right? Let's break it down step by step.

Understanding Arithmetic Series

Before we jump into the problem itself, let's quickly recap what an arithmetic series is. In simple terms, an arithmetic series is 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'. For example, the sequence 2, 5, 8, 11, 14... is an arithmetic series with a common difference of 3. Each term is obtained by adding 3 to the previous term.

Now, there are a couple of key formulas that we'll need to tackle this problem. First, the formula for the nth term (aₙ) of an arithmetic sequence:

aₙ = a₁ + (n - 1)d

Where:

• aₙ is the nth term
• a₁ is the first term
• n is the term number
• d is the common difference

This formula allows us to find any term in the sequence if we know the first term and the common difference. Pretty neat, huh?

Next, we have the formula for the sum of the first n terms (Sₙ) of an arithmetic series:

Sₙ = n/2 * [2a₁ + (n - 1)d]

Alternatively, we can also write this as:

Sₙ = n/2 * (a₁ + aₙ)

This formula tells us the sum of the first n terms if we know the number of terms, the first term, and the common difference (or the last term). These two formulas are our secret weapons for solving this problem!

Think of it this way: the sum of an arithmetic series is like finding the area of a trapezoid. The number of terms is like the height, the first term is one base, and the last term is the other base. The formula just puts this geometric intuition into a neat algebraic form. We can visualize the sum of the series as adding up these terms, and the formula gives us a shortcut to avoid adding them all individually. This is super useful when we have a large number of terms! So, with these formulas in our arsenal, we're ready to tackle the specific problem at hand.

Applying the Formulas to the Problem

In this problem, we're given that we have an arithmetic series with 12 terms (n = 12). We also know that the sum of the first 9 terms is 9 (S₉ = 9), and the sum of the last 3 terms is 63. Our goal is to find the sum of all 12 terms (S₁₂). It sounds like a puzzle, but with our handy formulas, we can solve it. The key is to use the information we have to find the first term (a₁) and the common difference (d). Once we have those, we can easily calculate the sum of all 12 terms.

First, let's use the information about the sum of the first 9 terms. We know that:

S₉ = 9

Using the formula for the sum of an arithmetic series, we can write:

9 = 9/2 * [2a₁ + (9 - 1)d]

Simplifying this equation, we get:

9 = 9/2 * (2a₁ + 8d)

Let's multiply both sides by 2/9 to get rid of the fraction:

2 = 2a₁ + 8d

We can further simplify this by dividing both sides by 2:

1 = a₁ + 4d

This gives us our first equation relating a₁ and d. Now, let's tackle the information about the sum of the last 3 terms. This is where things get a little trickier, but stay with me. The last 3 terms are the 10th, 11th, and 12th terms. Their sum is 63. So, we can write:

a₁₀ + a₁₁ + a₁₂ = 63

We can express each of these terms using the formula for the nth term:

(a₁ + 9d) + (a₁ + 10d) + (a₁ + 11d) = 63

Combining like terms, we get:

3a₁ + 30d = 63

Let's divide both sides by 3 to simplify:

a₁ + 10d = 21

This is our second equation relating a₁ and d. Now we have two equations with two unknowns, which is a classic algebra problem! We can use these two equations to solve for a₁ and d.

Solving the System of Equations

We have the following two equations:

1. a₁ + 4d = 1
2. a₁ + 10d = 21

We can use several methods to solve this system of equations, such as substitution or elimination. Let's use the elimination method. We can subtract the first equation from the second equation to eliminate a₁:

(a₁ + 10d) - (a₁ + 4d) = 21 - 1

Simplifying this, we get:

6d = 20

Dividing both sides by 6, we find the common difference:

d = 20/6 = 10/3

Now that we have the common difference, we can substitute it back into either equation to find the first term. Let's use the first equation:

a₁ + 4(10/3) = 1

a₁ + 40/3 = 1

Subtracting 40/3 from both sides, we get:

a₁ = 1 - 40/3 = 3/3 - 40/3 = -37/3

So, we've found that the first term is -37/3 and the common difference is 10/3. Now we're ready to find the sum of all 12 terms!

This part of the problem really showcases how math builds on itself. We used the formulas for arithmetic series, then we set up a system of linear equations, and now we're going to use those solutions to find the final answer. It's like a mathematical puzzle with interconnected pieces. The elimination method is particularly useful here because it's a straightforward way to get rid of one variable and solve for the other. Guys, remember that these techniques aren't just for arithmetic series; they pop up in all sorts of math and science problems. Mastering them is like leveling up your problem-solving skills!

Calculating the Sum of All 12 Terms

Now that we know the first term (a₁ = -37/3) and the common difference (d = 10/3), we can use the formula for the sum of an arithmetic series to find the sum of all 12 terms (S₁₂):

S₁₂ = 12/2 * [2a₁ + (12 - 1)d]

Substituting the values we found, we get:

S₁₂ = 6 * [2(-37/3) + 11(10/3)]

S₁₂ = 6 * [-74/3 + 110/3]

S₁₂ = 6 * [36/3]

S₁₂ = 6 * 12

S₁₂ = 72

Therefore, the sum of all 12 terms in the arithmetic series is 72. Awesome! We did it!

Guys, this final calculation really brings everything together. We started with the formulas, then we used the given information to set up equations, solved those equations, and finally plugged the results back into the sum formula. It's a beautiful chain of mathematical reasoning! Notice how the fractions might have looked intimidating at first, but by carefully following the steps and simplifying, we arrived at a clean integer answer. This is a common theme in math problems – don't be afraid of the messy bits, just keep going and trust the process. Each step, like simplifying the fractions or plugging the values into the formula, is a small victory that leads us closer to the final solution. And when we get there, it feels pretty darn good!

Conclusion

So, we successfully found the sum of all the terms in the arithmetic series. This problem demonstrates the power of understanding the properties of arithmetic series and using the right formulas. We also saw how solving a system of equations can be a valuable tool in tackling math problems. Remember, practice makes perfect! The more you work with these concepts, the more comfortable you'll become with them. And who knows, maybe you'll even start to enjoy them (math can be fun, I promise!). Keep exploring, keep learning, and keep solving!

To summarize:

• We started by understanding the key formulas for arithmetic series.
• We used the given information to set up a system of equations.
• We solved the system of equations to find the first term and the common difference.
• Finally, we used these values to calculate the sum of all 12 terms.

The final answer is 72.

I hope this explanation was helpful and clear. If you have any questions or want to explore more math problems, feel free to ask! Keep up the great work, everyone!

Key takeaways from this problem:

• Understand the Formulas: The formulas for the nth term and the sum of an arithmetic series are crucial. Memorize them and know how to apply them.
• Translate Words into Equations: The ability to translate word problems into mathematical equations is a key skill in algebra.
• Solve Systems of Equations: Mastering techniques like substitution and elimination is essential for solving many math problems.
• Don't Be Afraid of Fractions: Fractions can sometimes look scary, but with careful simplification, they become manageable.
• Practice Makes Perfect: The more you practice, the more confident you'll become in your problem-solving abilities.

Further Exploration:

If you enjoyed this problem, you might want to explore other types of series, such as geometric series or infinite series. You can also investigate different methods for solving systems of equations, such as using matrices. The world of mathematics is vast and fascinating, so keep exploring! And remember, every problem you solve makes you a stronger mathematician. So, guys, keep challenging yourselves and keep having fun with math!