Sum Of The First 57 Terms Arithmetic Sequence -17, -12, -7

Understanding Arithmetic Sequences and Series

Before we dive into calculating the sum of the first 57 terms of the given arithmetic sequence, let's first understand the fundamental concepts of arithmetic sequences and series. An arithmetic sequence is a sequence of numbers in which 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, 4, 6, 8, 10... is an arithmetic sequence with a common difference of 2. Each term is obtained by adding the common difference to the previous term. Understanding the definition of arithmetic sequences is essential for solving problems related to finding the sum of terms within such sequences.

An arithmetic series, on the other hand, is the sum of the terms in an arithmetic sequence. If we have an arithmetic sequence a₁, a₂, a₃, ..., aₙ, then the corresponding arithmetic series is a₁ + a₂ + a₃ + ... + aₙ. The sum of an arithmetic series can be calculated using specific formulas, which we will discuss in detail later. Recognizing the difference between a sequence and a series is crucial, as it dictates the approach we take in solving related problems. This article focuses on finding the sum of the first 57 terms, which falls under the domain of arithmetic series.

The beauty of arithmetic sequences and series lies in their predictable nature. The constant common difference allows us to easily determine any term in the sequence or calculate the sum of any number of terms. This predictability makes them a common topic in mathematics and has practical applications in various fields, such as finance and physics. In finance, for instance, simple interest calculations involve arithmetic sequences. In physics, uniformly accelerated motion can be modeled using arithmetic sequences. Thus, mastering the concepts of arithmetic sequences and series is not only academically beneficial but also practically useful.

Identifying the Given Information

In this specific problem, we are given an arithmetic sequence -17, -12, -7, ... and we are tasked with finding the sum of the first 57 terms. To solve this, we need to identify the key parameters provided: the first term (a₁), the number of terms (n), and the common difference (d). Correctly identifying these values is the first step towards applying the appropriate formula for calculating the sum of an arithmetic series. A misidentification at this stage can lead to an incorrect final answer, so meticulous attention to detail is paramount.

The first term (a₁) is the initial value of the sequence. In our case, a₁ is -17. This is the starting point from which the rest of the sequence is generated by repeatedly adding the common difference. The first term is a critical component in the formula for the sum of an arithmetic series, as it anchors the series and influences the magnitude of the sum. Often, the first term is explicitly stated in the problem, but in some cases, it may need to be deduced from other information provided. Understanding how to correctly identify the first term is vital for solving a wide range of arithmetic series problems.

The number of terms (n) refers to how many terms we are summing up in the series. Here, we are asked to find the sum of the first 57 terms, so n = 57. This value determines the extent of the series we are considering. The number of terms directly affects the sum; the more terms we add, the larger the sum (assuming the terms are not decreasing significantly). Recognizing the number of terms is straightforward when explicitly given, but sometimes, it might be embedded within the problem's context, requiring careful reading and interpretation. The value of 'n' is indispensable when applying the formulas for the sum of an arithmetic series.

The common difference (d) is the constant value added to each term to obtain the next term in the sequence. To find the common difference, we subtract any term from its subsequent term. In our sequence, -12 - (-17) = 5 and -7 - (-12) = 5, so the common difference d is 5. The common difference is the defining characteristic of an arithmetic sequence; it dictates the rate at which the sequence increases or decreases. A positive common difference indicates an increasing sequence, while a negative common difference indicates a decreasing sequence. Accurately calculating the common difference is essential for both finding specific terms in the sequence and calculating the sum of the series.

Thus, we have identified the given information as follows:

• a₁ = -17 (the first term)
• n = 57 (the number of terms)
• d = 5 (the common difference)

Applying the Arithmetic Series Formula

Now that we have identified the key parameters of our arithmetic sequence, we can move on to calculating the sum of the first 57 terms. The formula to calculate the sum (Sₙ) of the first n terms of an arithmetic series is given by: Sₙ = n/2 * [2a₁ + (n - 1)d]. This formula is derived from the fundamental properties of arithmetic sequences and provides an efficient way to find the sum without having to add each term individually. Mastering this formula is crucial for solving a wide range of arithmetic series problems. The formula elegantly encapsulates the relationship between the number of terms, the first term, the common difference, and the total sum of the series.

The formula Sₙ = n/2 * [2a₁ + (n - 1)d] can be understood as averaging the first and last terms and then multiplying by the number of terms. The term [2a₁ + (n - 1)d] represents twice the first term plus the common difference multiplied by one less than the number of terms, which effectively calculates the nth term (last term) of the sequence. By adding the first term to the last term, we get the sum of the first and last terms. Dividing this by 2 gives us the average term. Finally, multiplying the average term by the number of terms gives us the total sum. This interpretation provides an intuitive understanding of why the formula works.

To apply the formula, we substitute the values we identified earlier: a₁ = -17, n = 57, and d = 5. Substituting these values into the formula, we get: S₅₇ = 57/2 * [2(-17) + (57 - 1)5]. This substitution is a critical step, and it's important to ensure that the values are placed correctly in the formula to avoid errors. Once the values are substituted, we proceed with the arithmetic operations to simplify the expression and calculate the final sum. This process involves careful attention to the order of operations (PEMDAS/BODMAS) to ensure accuracy. The correct substitution and subsequent calculation are essential for arriving at the correct solution.

Calculating the Sum

Having substituted the values into the arithmetic series formula, we now proceed with the calculation to find the sum of the first 57 terms. The formula with the substituted values is: S₅₇ = 57/2 * [2(-17) + (57 - 1)5]. The first step is to simplify the expression inside the brackets, following the order of operations (PEMDAS/BODMAS). This involves performing the multiplication and subtraction within the brackets before moving on to the rest of the calculation. Precision in these calculations is vital, as any error here will propagate through the rest of the solution.

First, we calculate 2(-17) which equals -34. Next, we calculate (57 - 1) which equals 56. Then, we multiply 56 by 5, resulting in 280. So, the expression inside the brackets becomes -34 + 280. Adding these two values gives us 246. Therefore, our equation now looks like this: S₅₇ = 57/2 * 246. This simplification demonstrates the step-by-step approach to solving the problem, emphasizing the importance of breaking down complex calculations into manageable steps. Accuracy at each step is crucial for achieving the correct final answer.

Next, we divide 57 by 2, which gives us 28.5. Now, we multiply 28.5 by 246. This final multiplication will give us the sum of the first 57 terms of the arithmetic sequence. Performing this multiplication, we get: S₅₇ = 28.5 * 246 = 7011. Therefore, the sum of the first 57 terms of the arithmetic sequence -17, -12, -7, ... is 7011. This final calculation provides the solution to the problem and showcases the power of the arithmetic series formula in efficiently summing a large number of terms.

Therefore, S₅₇ = 7011.

Final Answer

In conclusion, the sum of the first 57 terms of the arithmetic sequence -17, -12, -7, ... is 7011. This result was obtained by first identifying the key parameters of the sequence: the first term (a₁ = -17), the number of terms (n = 57), and the common difference (d = 5). We then applied the arithmetic series formula, Sₙ = n/2 * [2a₁ + (n - 1)d], to calculate the sum. Substituting the identified values into the formula and performing the necessary arithmetic operations, we arrived at the final answer. This process demonstrates a methodical approach to solving arithmetic series problems, emphasizing the importance of accurately identifying given information and correctly applying the appropriate formula. The final answer, 7011, represents the total accumulation of adding the first 57 terms of the given sequence.

This problem exemplifies a common type of question encountered in mathematics, particularly in the study of sequences and series. The arithmetic series formula provides a powerful tool for efficiently calculating the sum of terms in an arithmetic sequence, avoiding the need for manual addition of each term. The ability to solve such problems is crucial for developing a strong foundation in mathematical concepts and has applications in various fields, including finance, physics, and computer science. Understanding and applying these concepts enhances problem-solving skills and mathematical reasoning abilities.

Moreover, this exercise highlights the importance of attention to detail and accuracy in mathematical calculations. Each step, from identifying the given parameters to substituting them into the formula and performing the arithmetic operations, requires careful attention to avoid errors. A small mistake at any stage can lead to an incorrect final answer. Therefore, practicing and mastering these skills is essential for success in mathematics and related disciplines. The result, 7011, not only answers the specific question but also reinforces the broader principles of mathematical problem-solving.