Finding The Sum Of The First 30 Terms Of The Sequence A_n = 6n + 5

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Understanding arithmetic sequences is crucial in mathematics, and calculating the sum of a finite number of terms is a fundamental skill. In this article, we will delve into how to find the sum of the first 30 terms of the arithmetic sequence defined by the formula a_n = 6n + 5. This comprehensive guide will walk you through the process step by step, ensuring you grasp the underlying concepts and can apply them to similar problems.

Understanding 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 called the common difference, often denoted by 'd'. The general form of an arithmetic sequence is:

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

where 'a' is the first term and 'd' is the common difference. In our case, the sequence is defined by a_n = 6n + 5. To confirm this is an arithmetic sequence, we can find the first few terms and check for a common difference.

Let's calculate the first few terms:

  • For n = 1, a_1 = 6(1) + 5 = 11
  • For n = 2, a_2 = 6(2) + 5 = 17
  • For n = 3, a_3 = 6(3) + 5 = 23
  • For n = 4, a_4 = 6(4) + 5 = 29

The sequence begins as 11, 17, 23, 29, ... The difference between consecutive terms is:

  • 17 - 11 = 6
  • 23 - 17 = 6
  • 29 - 23 = 6

Since the difference between consecutive terms is consistently 6, we can confirm that this is indeed an arithmetic sequence with a common difference of 6. Recognizing that the sequence is arithmetic is the first step in efficiently calculating the sum of its terms.

Formula for the Sum of an Arithmetic Series

The sum of the first 'n' terms of an arithmetic sequence, often denoted as S_n, can be calculated using the following formula:

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

Where:

  • S_n is the sum of the first 'n' terms.
  • 'n' is the number of terms.
  • 'a' is the first term of the sequence.
  • 'd' is the common difference.

This formula is derived from the properties of arithmetic sequences and provides a straightforward method to find the sum without having to add each term individually. Another useful form of this formula, which can be particularly helpful when the last term is known, is:

S_n = n/2 * (a + l)

Where:

  • 'l' is the last term of the sequence (the nth term).

This alternative formula is essentially averaging the first and last terms and multiplying by the number of terms. Understanding these formulas is key to solving problems related to arithmetic series efficiently.

Applying the Formula to Our Problem

Now that we understand the formula for the sum of an arithmetic series, let's apply it to our specific problem. We are asked to find the sum of the first 30 terms of the sequence a_n = 6n + 5. We have already determined that this is an arithmetic sequence with:

  • First term (a) = 11
  • Common difference (d) = 6
  • Number of terms (n) = 30

Using the formula S_n = n/2 * [2a + (n - 1)d], we can substitute these values:

S_30 = 30/2 * [2(11) + (30 - 1)6]

Now, let's simplify the expression:

S_30 = 15 * [22 + (29)6] S_30 = 15 * [22 + 174] S_30 = 15 * 196 S_30 = 2940

Therefore, the sum of the first 30 terms of the sequence a_n = 6n + 5 is 2940. This methodical application of the formula demonstrates the power of understanding arithmetic sequences and their properties.

Step-by-Step Calculation

To further clarify the calculation, let's break down each step in detail. This step-by-step approach will reinforce the process and ensure clarity.

  1. Identify the values:
    • n = 30 (number of terms)
    • a = 11 (first term)
    • d = 6 (common difference)
  2. Write down the formula:
    • S_n = n/2 * [2a + (n - 1)d]
  3. Substitute the values into the formula:
    • S_30 = 30/2 * [2(11) + (30 - 1)6]
  4. Simplify inside the brackets:
    • S_30 = 15 * [22 + (29)6]
  5. Calculate the product inside the brackets:
    • S_30 = 15 * [22 + 174]
  6. Add the numbers inside the brackets:
    • S_30 = 15 * 196
  7. Multiply to find the sum:
    • S_30 = 2940

By breaking down the calculation into these steps, we can see how each value contributes to the final sum. This detailed approach is useful for checking your work and ensuring accuracy.

Alternative Method Using the Last Term

As mentioned earlier, there is an alternative formula to calculate the sum if we know the last term (l). Let's use this method to verify our result. First, we need to find the 30th term of the sequence:

  • a_30 = 6(30) + 5 = 180 + 5 = 185

So, the last term (l) is 185. Now, we can use the formula:

S_n = n/2 * (a + l)

Substitute the values:

S_30 = 30/2 * (11 + 185) S_30 = 15 * 196 S_30 = 2940

As we can see, this alternative method gives us the same result, 2940, confirming our previous calculation. Using different methods to solve the same problem is a good practice to ensure accuracy and deepen understanding.

Common Mistakes and How to Avoid Them

When calculating the sum of an arithmetic series, several common mistakes can occur. Recognizing these mistakes and understanding how to avoid them is essential for accurate problem-solving.

  1. Incorrectly identifying the first term (a) or common difference (d): This is a fundamental error that will propagate through the entire calculation. Always double-check the sequence and ensure you have correctly identified 'a' and 'd'. For instance, if the sequence is given by a formula like a_n = 6n + 5, make sure to calculate a_1 correctly by substituting n = 1. Similarly, the common difference should be verified by subtracting consecutive terms.

  2. Using the wrong formula: It's crucial to use the correct formula for the sum of an arithmetic series. Confusing it with the formula for the nth term or a geometric series will lead to an incorrect answer. Remember, the sum formula is S_n = n/2 * [2a + (n - 1)d] or S_n = n/2 * (a + l).

  3. Arithmetic errors: Simple arithmetic mistakes, such as incorrect multiplication or addition, can easily occur. It's always a good idea to perform calculations carefully and, if possible, use a calculator to double-check your work. Pay close attention to the order of operations (PEMDAS/BODMAS) when simplifying expressions.

  4. Misunderstanding the number of terms (n): Ensure you correctly identify the number of terms you are summing. If the question asks for the sum of the first 30 terms, then n = 30. However, if the question asks for the sum of terms from, say, the 5th to the 30th, you need to calculate the correct number of terms (which would be 26 in this case).

  5. Forgetting to divide by 2 in the formula: The factor of n/2 in the sum formulas is crucial. Forgetting to divide by 2 is a common mistake that will lead to a significantly incorrect answer. Always double-check that you have included this factor in your calculation.

To avoid these mistakes, it's helpful to:

  • Write down the formula: Before substituting values, write down the formula to ensure you are using the correct one.
  • Double-check your values: Verify that you have correctly identified 'a', 'd', and 'n'.
  • Perform calculations carefully: Pay attention to arithmetic and the order of operations.
  • Use an alternative method: If possible, use a different method to solve the same problem and check your answer.
  • Review your work: After completing the calculation, review each step to catch any potential errors.

Real-World Applications of Arithmetic Sequences

While arithmetic sequences may seem like a purely mathematical concept, they have numerous real-world applications. Understanding these applications can help appreciate the practical relevance of this topic.

  1. Simple Interest: Calculating simple interest on a loan or investment involves arithmetic sequences. The interest earned each year is a constant amount, forming an arithmetic sequence. The total amount after a certain number of years can be calculated using the sum of the arithmetic series.

  2. Salary Increments: Many jobs offer annual salary increments that form an arithmetic sequence. For example, if a person starts with a salary of $50,000 and receives a $2,000 increment each year, their salary over the years forms an arithmetic sequence. Calculating their total earnings over a period of time can involve summing the series.

  3. Depreciation: The value of some assets, like machinery or vehicles, depreciates over time in a linear fashion, forming an arithmetic sequence. Understanding this depreciation can help in financial planning and asset management.

  4. Stacking Objects: When stacking objects in a uniform manner, such as logs or chairs, the number of objects in each layer can form an arithmetic sequence. Calculating the total number of objects requires summing the series.

  5. Patterns in Nature: Some patterns in nature, such as the arrangement of seeds in a sunflower or the branching of trees, can exhibit arithmetic sequences. While not always perfect, these patterns can be modeled and understood using mathematical concepts.

  6. Construction and Engineering: In construction, tasks like laying bricks in rows or placing pipes can involve arithmetic sequences. Calculating the total number of bricks or pipes needed often requires summing an arithmetic series.

  7. Financial Planning: Saving money regularly with fixed increments forms an arithmetic sequence. Calculating the total savings after a certain period can be done using the sum of the series. This is useful in retirement planning or other long-term financial goals.

By understanding these real-world applications, the importance of learning about arithmetic sequences becomes clear. It's not just an abstract mathematical concept but a tool that can be used to solve practical problems in various fields.

Conclusion

In this article, we have comprehensively explored how to find the sum of the first 30 terms of the sequence a_n = 6n + 5. We began by understanding the definition of an arithmetic sequence and verifying that the given sequence is indeed arithmetic. We then introduced the formula for the sum of an arithmetic series, S_n = n/2 * [2a + (n - 1)d], and applied it step by step to our problem, arriving at the solution of 2940. We also explored an alternative method using the last term and discussed common mistakes to avoid. Finally, we highlighted the real-world applications of arithmetic sequences, demonstrating their practical relevance.

By mastering the concepts and techniques discussed in this article, you will be well-equipped to tackle similar problems involving arithmetic sequences and series. Remember, practice is key to success in mathematics, so continue to solve problems and deepen your understanding.