Arithmetic Sequence Sum Find The Sum Of First Six Terms Of 2 7 12 Sequence

In the realm of mathematics, arithmetic sequences hold a fundamental position, characterized by a consistent difference between consecutive terms. These sequences appear in various real-world scenarios, from simple counting patterns to complex financial models. Understanding how to manipulate and calculate with arithmetic sequences is a crucial skill for students and professionals alike. In this article, we will delve into the specifics of finding the sum of the first six terms of a particular arithmetic sequence: 2, 7, 12, .... This exploration will not only provide a step-by-step solution to this problem but also illuminate the underlying principles and formulas that govern arithmetic sequences.

Understanding Arithmetic Sequences

Before diving into the problem, it's crucial to establish a solid understanding of arithmetic sequences. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, often denoted as 'd'. The general form of an arithmetic sequence can be written as:

a, a + d, a + 2d, a + 3d, ...

where 'a' represents the first term of the sequence. The nth term of an arithmetic sequence, denoted as a_n, can be found using the formula:

a_n = a + (n - 1)d

This formula allows us to calculate any term in the sequence without having to list out all the preceding terms. Understanding this formula is paramount to tackling more complex problems related to arithmetic sequences. In our case, the sequence 2, 7, 12, ... provides a clear example of an arithmetic sequence. The first term, 'a', is 2. To find the common difference, 'd', we subtract the first term from the second term (or the second from the third, and so on): 7 - 2 = 5. So, the common difference, 'd', is 5. This means that each term in the sequence is 5 greater than the previous term. Recognizing these fundamental characteristics of an arithmetic sequence sets the stage for calculating the sum of its terms.

Identifying the Sequence and Its Properties

Our focus is on the arithmetic sequence 2, 7, 12, .... To effectively find the sum of the first six terms, we first need to clearly identify the key properties of this sequence. As we established earlier, the first term, denoted as 'a', is 2. The common difference, 'd', which is the constant difference between consecutive terms, is calculated as 7 - 2 = 5. Thus, d = 5. Now, we want to find the sum of the first six terms, which means we are interested in the first six numbers in this sequence. These terms are generated by repeatedly adding the common difference to the previous term. So, the first six terms are: 2, 7, 12, 17, 22, and 27. To verify, we can also use the formula for the nth term, a_n = a + (n - 1)d, to find the sixth term. Plugging in a = 2, d = 5, and n = 6, we get: a_6 = 2 + (6 - 1) * 5 = 2 + 5 * 5 = 2 + 25 = 27. This confirms that our sequence is correctly identified. Having these terms explicitly listed allows us to visualize the pattern and provides a direct way to calculate the sum by simply adding them together. However, for larger sequences, this method can be tedious. This is where the formula for the sum of an arithmetic series becomes invaluable, offering a more efficient approach to finding the sum.

Applying the Arithmetic Series Formula

The most efficient method for finding the sum of an arithmetic sequence involves using the arithmetic series formula. This formula provides a direct way to calculate the sum without having to add each term individually. The formula for the sum of the first 'n' terms of an arithmetic series, denoted as S_n, is given by:

S_n = n/2 * [2a + (n - 1)d]

where 'n' is the number of terms, 'a' is the first term, and 'd' is the common difference. This formula is derived from the concept of pairing the first and last terms, the second and second-to-last terms, and so on, each pair summing to the same value. In our specific problem, we want to find the sum of the first six terms, so n = 6. We already know that the first term, a, is 2 and the common difference, d, is 5. Now, we can substitute these values into the formula: S_6 = 6/2 * [2 * 2 + (6 - 1) * 5] This simplifies to: S_6 = 3 * [4 + 5 * 5] S_6 = 3 * [4 + 25] S_6 = 3 * 29 S_6 = 87 Therefore, the sum of the first six terms of the arithmetic sequence 2, 7, 12, ... is 87. This formula not only provides the correct answer but also demonstrates the power of mathematical formulas in simplifying complex calculations. By understanding and applying this formula, we can efficiently calculate the sum of any arithmetic series, regardless of its length.

Step-by-Step Calculation of the Sum

To solidify our understanding, let's walk through the step-by-step calculation of the sum of the first six terms of the arithmetic sequence 2, 7, 12, ... using the arithmetic series formula. 1. Identify the variables: First, we need to identify the key variables in our problem. We know that the number of terms, n, is 6. The first term, a, is 2. The common difference, d, is 5. 2. Write down the formula: The formula for the sum of an arithmetic series is: S_n = n/2 * [2a + (n - 1)d] 3. Substitute the values: Substitute the identified values into the formula: S_6 = 6/2 * [2 * 2 + (6 - 1) * 5] 4. Simplify inside the brackets: Simplify the expression inside the brackets first, following the order of operations (PEMDAS/BODMAS): S_6 = 3 * [4 + (5 * 5)] S_6 = 3 * [4 + 25] 5. Continue simplifying: Add the numbers inside the brackets: S_6 = 3 * 29 6. Multiply: Finally, multiply the result by 3: S_6 = 87 Therefore, the sum of the first six terms of the arithmetic sequence 2, 7, 12, ... is 87. This step-by-step approach clearly illustrates how the formula is applied and provides a clear path for solving similar problems. By breaking down the calculation into manageable steps, we minimize the chance of errors and ensure a solid understanding of the process.

Verifying the Result by Direct Addition

While the arithmetic series formula offers an efficient way to calculate the sum, it's always a good practice to verify the result, especially when dealing with mathematical problems. One way to verify our answer is by directly adding the first six terms of the sequence. We've already identified the first six terms as 2, 7, 12, 17, 22, and 27. Now, let's add them together: 2 + 7 + 12 + 17 + 22 + 27 = 87 As we can see, the sum obtained by direct addition matches the result we obtained using the arithmetic series formula. This verification step not only confirms the accuracy of our calculations but also reinforces our understanding of the arithmetic sequence and its properties. This method of direct addition serves as a valuable check, especially when dealing with a limited number of terms. It provides a tangible understanding of what the formula represents – the sum of the terms in the sequence. Furthermore, this verification process underscores the importance of accuracy in mathematical calculations. By employing different methods to arrive at the same answer, we strengthen our confidence in the solution and the process itself.

Practical Applications of Arithmetic Sequences

Arithmetic sequences are not just abstract mathematical concepts; they have numerous practical applications in various real-world scenarios. Understanding arithmetic sequences can be incredibly beneficial in fields such as finance, physics, computer science, and even everyday life. One common application is in finance, particularly in calculating simple interest. For example, if you deposit a certain amount of money into an account that earns simple interest, the balance of the account will grow as an arithmetic sequence. Each term in the sequence represents the balance after a certain period, with the common difference being the interest earned in each period. In physics, arithmetic sequences can be used to model situations involving constant acceleration. For instance, if an object is accelerating at a constant rate, the distance it travels in equal time intervals forms an arithmetic sequence. The terms of the sequence represent the distances traveled in each interval, and the common difference represents the increase in distance due to constant acceleration. In computer science, arithmetic sequences are used in algorithms and data structures. For example, accessing elements in an array with a fixed stride can be viewed as an arithmetic sequence. Understanding the properties of arithmetic sequences can help optimize these algorithms. Beyond these specific fields, arithmetic sequences can also help in everyday problem-solving. For example, if you're planning a workout routine and want to gradually increase the number of repetitions you do each week, you can use an arithmetic sequence to determine your progress. The applications of arithmetic sequences are vast and varied, highlighting their importance as a fundamental mathematical concept. By understanding these sequences, we can better model and solve problems in a wide range of situations.

Conclusion Mastering Arithmetic Sequence Sums

In conclusion, finding the sum of the first six terms of the arithmetic sequence 2, 7, 12, ... is a problem that not only demonstrates the application of the arithmetic series formula but also underscores the importance of understanding the fundamental properties of arithmetic sequences. We successfully calculated the sum to be 87 using the formula S_n = n/2 * [2a + (n - 1)d], and we verified this result through direct addition, reinforcing the accuracy of our solution. This exploration has highlighted the step-by-step process involved in applying the arithmetic series formula, emphasizing the need to correctly identify the key variables: the number of terms, the first term, and the common difference. Furthermore, we have explored the practical applications of arithmetic sequences in various fields, illustrating their relevance beyond the classroom. From finance to physics to computer science, arithmetic sequences provide a powerful tool for modeling and solving real-world problems. Mastering the concepts and techniques discussed in this article will not only enhance your mathematical skills but also equip you with a valuable problem-solving approach applicable to a wide range of situations. Whether you are a student learning about sequences for the first time or a professional seeking to apply these concepts in your work, a solid understanding of arithmetic sequences is an invaluable asset.