Calculating The Sum Of The Arithmetic Sequence 4, 11 Up To The 11th Term
Understanding Arithmetic Sequences
In mathematics, an arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is known as the common difference. Arithmetic sequences are fundamental in various mathematical fields, including algebra and calculus, and understanding them is crucial for problem-solving in many real-world applications. To effectively calculate the sum of an arithmetic sequence, one must first grasp the core concepts that define it. Let's delve deeper into what constitutes an arithmetic sequence and how its properties enable us to efficiently compute sums.
To begin, consider the basic structure of an arithmetic sequence. Each term in the sequence is generated by adding the common difference to the preceding term. This creates a linear progression, where the terms increase or decrease by a fixed amount. Identifying the common difference is the first step in analyzing and working with arithmetic sequences. This difference allows us to predict future terms and calculate sums over a range of terms. For instance, if we know the first term and the common difference, we can determine any term in the sequence without having to list out all the preceding terms.
Moreover, the formula for the nth term of an arithmetic sequence is a cornerstone for solving related problems. The formula is expressed as 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, and d is the common difference. This formula allows us to directly calculate any term in the sequence, making it a powerful tool for various mathematical manipulations. Understanding how to use this formula is essential for tackling more complex problems involving arithmetic sequences. For example, we can use it to find a specific term in the sequence or to determine if a particular number belongs to the sequence.
Furthermore, the sum of an arithmetic sequence can be calculated using a specific formula, which we will explore in detail in the following sections. This formula leverages the properties of arithmetic sequences to provide a shortcut for summing terms without having to add them individually. By understanding the derivation and application of this formula, we can efficiently solve problems involving the summation of arithmetic series, saving time and reducing the risk of errors. The ability to calculate these sums is particularly useful in scenarios where dealing with large numbers of terms is necessary.
Identifying the Arithmetic Sequence and Its Parameters
In this specific problem, we are tasked with calculating the sum of the arithmetic sequence 4, 11... up to the 11th term. To solve this, the first crucial step involves identifying the key parameters of the sequence. These parameters include the first term (a_1), the common difference (d), and the number of terms (n) we need to consider. Accurately determining these values is essential for applying the appropriate formula and arriving at the correct answer. Misidentifying any of these parameters can lead to significant errors in the final calculation.
The first term, denoted as a_1, is the initial value in the sequence. In our case, the arithmetic sequence begins with 4, making a_1 = 4. This is the starting point from which all subsequent terms are generated by adding the common difference. The first term serves as the foundation for the sequence, and its value is critical for both understanding the sequence's progression and performing calculations. It is important to carefully observe the given sequence to correctly identify the first term.
Next, we need to determine the common difference (d). The common difference 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. For instance, in the sequence 4, 11..., we can subtract 4 from 11 to find the common difference: d = 11 - 4 = 7. This means that each term in the sequence is 7 greater than the previous term. The common difference is a defining characteristic of arithmetic sequences, and its correct calculation is vital for subsequent steps.
Lastly, we need to identify the number of terms (n) we are summing. The problem explicitly states that we need to calculate the sum up to the 11th term. Therefore, n = 11. This parameter tells us the extent of the sequence we need to consider when calculating the sum. Knowing the number of terms is crucial for using the summation formula accurately. If we were summing a different number of terms, the final result would change accordingly. With all three parameters (a_1, d, and n) identified, we are now well-prepared to apply the formula for the sum of an arithmetic sequence.
Applying the Arithmetic Series Sum Formula
Now that we have identified the key parameters of the arithmetic sequence – the first term (a_1 = 4), the common difference (d = 7), and the number of terms (n = 11) – we can proceed to apply the formula for the sum of an arithmetic series. This formula provides a direct method for calculating the sum of the first n terms of an arithmetic sequence, without the need to individually add each term. The formula is expressed as:
S_n = n/2 * [2a_1 + (n - 1)d]
Where:
- S_n is the sum of the first n terms.
- n is the number of terms.
- a_1 is the first term.
- d is the common difference.
This formula is derived from the average of the first and last terms, multiplied by the number of terms. It is a powerful tool that simplifies the calculation of arithmetic series sums, particularly when dealing with a large number of terms. Understanding how to correctly apply this formula is essential for solving problems efficiently and accurately.
To apply the formula to our specific problem, we substitute the values we identified earlier:
- a_1 = 4
- d = 7
- n = 11
Plugging these values into the formula, we get:
S_11 = 11/2 * [2(4) + (11 - 1)7]
Now, we perform the calculations step by step. First, we simplify the expression inside the brackets:
2(4) = 8 (11 - 1)7 = 10 * 7 = 70
So, the expression inside the brackets becomes:
8 + 70 = 78
Now, we substitute this back into the formula:
S_11 = 11/2 * 78
Next, we multiply 11/2 by 78:
S_11 = 5.5 * 78
Finally, we calculate the product:
S_11 = 429
Thus, the sum of the arithmetic sequence 4, 11... up to the 11th term is 429. This result provides a concrete solution to the problem, showcasing the effectiveness of the arithmetic series sum formula.
Step-by-Step Calculation and Solution
To ensure clarity and understanding, let’s walk through the step-by-step calculation to find the sum of the arithmetic sequence 4, 11... up to the 11th term. This detailed approach will not only provide the solution but also reinforce the application of the arithmetic series sum formula. By breaking down the problem into manageable steps, we can minimize errors and gain confidence in our calculations.
Step 1: Identify the First Term (a_1)
The first term of the sequence is the initial value. In the sequence 4, 11..., the first term is clearly 4. Therefore:
a_1 = 4
This is the foundation upon which the rest of the sequence is built. Correctly identifying the first term is crucial for subsequent calculations.
Step 2: Determine the Common Difference (d)
The common difference is the constant value added to each term to obtain the next term. To find this, we subtract any term from its subsequent term. Using the given sequence 4, 11..., we subtract 4 from 11:
d = 11 - 4 = 7
Thus, the common difference is 7. This means that each term in the sequence increases by 7 compared to the previous term.
Step 3: Identify the Number of Terms (n)
The problem specifies that we need to calculate the sum up to the 11th term. Therefore, the number of terms we are considering is 11:
n = 11
This value indicates the extent of the sequence we are summing, which is essential for using the summation formula accurately.
Step 4: Apply the Arithmetic Series Sum Formula
The formula for the sum of the first n terms of an arithmetic sequence is:
S_n = n/2 * [2a_1 + (n - 1)d]
We now substitute the values we identified earlier into this formula:
S_11 = 11/2 * [2(4) + (11 - 1)7]
Step 5: Simplify the Expression
First, we simplify the terms inside the brackets:
2(4) = 8 (11 - 1)7 = 10 * 7 = 70
So, the expression inside the brackets becomes:
8 + 70 = 78
Now, we substitute this back into the formula:
S_11 = 11/2 * 78
Step 6: Calculate the Sum
Next, we multiply 11/2 by 78:
S_11 = 5.5 * 78
Finally, we perform the multiplication:
S_11 = 429
Step 7: State the Solution
Therefore, the sum of the arithmetic sequence 4, 11... up to the 11th term is 429.
Conclusion and Answer
In conclusion, by carefully identifying the parameters of the arithmetic sequence and applying the arithmetic series sum formula, we have successfully calculated the sum of the sequence 4, 11... up to the 11th term. The step-by-step calculation ensured accuracy and clarity in our solution. Understanding and applying these mathematical concepts is essential for solving a wide range of problems in mathematics and related fields. The final answer to our problem is:
The sum of the arithmetic sequence 4, 11... up to the 11th term is 429.
This process demonstrates the power and efficiency of using formulas in mathematics to solve complex problems. By breaking down the problem into manageable steps and applying the appropriate formula, we arrived at the correct solution with confidence. The ability to work with arithmetic sequences and series is a fundamental skill in mathematics, and mastering these concepts will provide a solid foundation for further mathematical studies.
