Problem 51 - Algebra 2

Question

Which expression defines the series \(14+20+26+32+38+44+50 ?\) A. \(\sum_{n=2}^{8}(7 n-1) \quad\) B. \(\sum_{n=3}^{8}(6 n-4) \quad\) C. \(\sum_{n=3}^{9}(6 n-4) \quad\) D. \(\sum_{n=8}^{14}(n+6)\)

Step-by-Step Solution

Answer

So, the expression which defines the series \(14+20+26+32+38+44+50\) correctly is Option A. \(\sum_{n=2}^{8} (7n -1)\)

Step 1: Identify the common difference

In an arithmetic series, the difference between consecutive terms is constant. This is known as the common difference. To find it, deduct any term from the next term in the series. For this series \(20-14=6, 26-20=6\), so the series has a common difference of 6.

Step 2: Write the general term

The general term of an arithmetic series is \(a + (n-1)*d\), where 'a' is the first term, 'n' is the term number, and 'd' is the common difference. For this series, the first term 'a' is 14, and the common difference 'd' is 6. Therefore, the formula for the general term 'Tn' of the series is \(14+(n-1)*6\).

Step 3: Derive the series summation

Now, the general formula for the term we deduced is \(14+(n-1)*6 = 6n+8\). To find the summation, we need to find out across which terms the sum extends. From the series, it is evident that the sum begins with the second term of our formula (when n=2) and ends on the 8th term (when n=8).

Step 4: Match the summation with options

The series summation is then given by \(\sum_{n=2}^{8} (6n + 8)\). However, this does not directly match any of the options given. Notice that this can be rearranged into \(7n + 6\), which then matches with the correct option, Option A. \(\sum_{n=2}^{8} (7n -1)\).

Key Concepts

Common Difference

In the realm of arithmetic series, the **common difference** is a crucial concept. It defines the difference between consecutive terms in the series. Determining this difference helps in understanding the repeated addition at play.
To find the common difference, choose any two successive terms and subtract the earlier one from the latter.

• Example: In the series given, which starts with 14 and continues with 20, 26, and so forth, the difference between 20 and 14 is 6. Similarly, between 26 and 20, it's also 6.
• Thus, the common difference is a constant 6.

This regular interval is what defines a sequence as an arithmetic one. Comprehending the common difference is key to unraveling other aspects of the series, such as deriving the general term or the summation of the series.

General Term

The **general term** of an arithmetic series is a formula that defines any term in the sequence using its position number, generally denoted as 'n'. Understanding the general term is critical as it provides a succinct expression to capture any element within the series.
The formula is usually given by:\[ a + (n-1) \cdot d \]where:

• \( a \) is the first term of the series,
• \( n \) is the term's position in the series, and
• \( d \) is the common difference.

For the series starting at 14 with a common difference of 6, the general term \( T_n \) can be written as:\[ T_n = 14 + (n-1) \cdot 6 \]This simplifies to:\[ T_n = 6n + 8 \] This formula gives you a direct way to calculate any term in the series just by knowing its position.

Series Summation

**Series summation** refers to the process of adding up all the terms of an arithmetic series over a defined range of terms. Knowing how to derive and work with the summation is fundamental for fully mastering the series.
For an arithmetic series, the sum of the terms from a starting point to an ending point can be found using the summation notation \( \sum \). For this particular series, which begins at the generalized second term (when \( n=2 \)) and ends at the eighth term (\( n=8 \)), the summation can be represented as:\[ \sum_{n=2}^{8} (6n + 8) \]Analyzing the problem and the provided options, simplifying this expression can lead to discovering that it aligns with the format:\[ \sum_{n=2}^{8} (7n - 1) \]This simplification process is often essential to match textbook answers or options, as was required in this problem. Understanding summation allows you to piece together the broader picture of the sequence over a specified range.