Finding The Sum Of An Arithmetic Series A Step-by-Step Guide

Hey guys! Ever stumbled upon a seemingly complex math problem that just makes your head spin? Well, today we're diving deep into one such problem, breaking it down piece by piece until it's as clear as day. We're tackling the sum of the first 34 numbers in the arithmetic series:

$$147 + 130 + 113 + 96 + \ldots$$

And trust me, by the end of this article, you'll be a pro at solving these types of problems. Let's get started!

Understanding Arithmetic Series

Before we jump into solving the problem, let's make sure we're all on the same page about what an arithmetic series actually is. In essence, an arithmetic series is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is what we call the common difference, often denoted as 'd'.

In our series, 147, 130, 113, 96, ..., we can quickly figure out the common difference by subtracting any term from its subsequent term. For example:

$$130 - 147 = -17$$

$$113 - 130 = -17$$

$$96 - 113 = -17$$

So, our common difference, d, is -17. This tells us that each term in the series is 17 less than the previous term. Recognizing this pattern is the first key step in cracking the problem.

Another crucial element of an arithmetic series is the first term, which we usually denote as 'a_1'. In our case, the first term is 147. Now that we know the first term (a_1 = 147) and the common difference (d = -17), we're well-equipped to find any term in the series and, more importantly, the sum of the first 34 terms.

The Nth Term Formula

To find the sum of the first 34 terms, we first need to know what the 34th term actually is. This is where the formula for the nth term of an arithmetic sequence comes in handy. The formula is:

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

Where:

• a_n is the nth term
• a_1 is the first term
• n is the term number
• d is the common difference

Let's plug in our values to find the 34th term (a_34):

$$a_{34} = 147 + (34 - 1)(-17)$$

$$a_{34} = 147 + (33)(-17)$$

$$a_{34} = 147 - 561$$

$$a_{34} = -414$$

So, the 34th term in the series is -414. This is a significant piece of the puzzle, and now we're ready to move on to the final step: calculating the sum.

Calculating the Sum of an Arithmetic Series

Now comes the exciting part where we actually calculate the sum of the first 34 terms. For this, we'll use the formula for the sum of an arithmetic series, which is:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

Where:

• S_n is the sum of the first n terms
• n is the number of terms
• a_1 is the first term
• a_n is the nth term

We know that n = 34, a_1 = 147, and we just calculated a_34 = -414. Let's plug these values into the formula:

$$S_{34} = \frac{34}{2}(147 + (-414))$$

$$S_{34} = 17(147 - 414)$$

$$S_{34} = 17(-267)$$

$$S_{34} = -4539$$

Therefore, the sum of the first 34 numbers in the series is -4539. Boom! We've cracked it.

Step-by-Step Solution Breakdown

Let's recap the steps we took to solve this problem. Breaking it down like this can help solidify your understanding and make tackling similar problems a breeze:

1. Identify the series as arithmetic: We recognized that the difference between consecutive terms was constant, indicating an arithmetic series.
2. Find the common difference (d): We calculated d by subtracting a term from its subsequent term, finding d = -17.
3. Identify the first term (a_1): We noted that the first term, a_1, is 147.
4. Find the nth term (a_n): Using the formula a_n = a_1 + (n - 1)d, we calculated the 34th term (a_34) to be -414.
5. Calculate the sum (S_n): Using the formula S_n = (n/2)(a_1 + a_n), we found the sum of the first 34 terms (S_34) to be -4539.

By following these steps, you can confidently tackle any arithmetic series sum problem that comes your way.

Common Pitfalls and How to Avoid Them

Math problems, especially those involving series, can sometimes be tricky. Here are a few common pitfalls to watch out for and how to avoid them:

• Misidentifying the series type: It's crucial to correctly identify whether a series is arithmetic, geometric, or neither. Confusing them can lead to using the wrong formulas and getting incorrect answers. Always check the differences or ratios between terms to confirm the series type.
• Incorrectly calculating the common difference: A simple mistake in subtraction can throw off the entire calculation. Double-check your subtraction and ensure you're subtracting in the correct order (subsequent term minus preceding term).
• Using the wrong formula: Using the formula for the nth term when you need the sum formula (or vice versa) is a common error. Make sure you understand what the problem is asking and choose the appropriate formula.
• Arithmetic errors: Even if you know the formulas and the steps, a simple arithmetic mistake can lead to a wrong answer. Take your time, double-check your calculations, and use a calculator if needed.
• Forgetting the negative sign: Especially with negative common differences or terms, it's easy to drop a negative sign. Pay close attention to signs and make sure you're carrying them through your calculations correctly.

By being mindful of these potential pitfalls, you can minimize errors and increase your accuracy in solving arithmetic series problems.

Real-World Applications of Arithmetic Series

Now, you might be wondering,