Find The 7th Term Of The Arithmetic Sequence 4x + 5, 11x + 9,18x + 13, ...

Find The 7th Term Of The Arithmetic Sequence 4x + 5, 11x + 9, 18x + 13, ...
Understanding how to find specific terms in an arithmetic sequence is a fundamental aspect of algebra and mathematical sequences. In this article, we will explore the process step-by-step to determine the 7th term of the given sequence: 4x + 5, 11x + 9, 18x + 13, ... We will analyze the pattern, derive the general term, and then calculate the 7th term, all while providing clear explanations to enhance your comprehension of arithmetic sequences.

What Is an Arithmetic Sequence?

Before diving into the specific sequence, it's essential to understand what an arithmetic sequence is.

Definition of an Arithmetic Sequence

An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a fixed number, called the common difference, to the previous term. Mathematically, an arithmetic sequence can be expressed as: \[ an = a1 + (n - 1)d \] where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term,
• \( d \) is the common difference,
• \( n \) is the position of the term in the sequence.

Analyzing the Given Sequence

The sequence provided is: \[ 4x + 5, \quad 11x + 9, \quad 18x + 13, \quad ... \]

Let's analyze the pattern to identify the common difference and the general term.

Step 1: Identify the First Few Terms

• First term: \( a_1 = 4x + 5 \)
• Second term: \( a_2 = 11x + 9 \)
• Third term: \( a_3 = 18x + 13 \)

Step 2: Find the Differences Between Terms

Calculate the difference between consecutive terms: \[ a2 - a1 = (11x + 9) - (4x + 5) = (11x - 4x) + (9 - 5) = 7x + 4 \] \[ a3 - a2 = (18x + 13) - (11x + 9) = (18x - 11x) + (13 - 9) = 7x + 4 \]

Since the differences are equal, the sequence is indeed arithmetic with common difference:
\[ d = 7x + 4 \]

Deriving the General Term

Now that we've identified the first term and the common difference, we can formulate the general term \( a_n \).

Step 1: Recall the Arithmetic Sequence Formula

\[ an = a1 + (n - 1)d \]

Step 2: Substitute Known Values

\[ a_n = (4x + 5) + (n - 1)(7x + 4) \]

Step 3: Simplify the Expression

Expand the expression: \[ a_n = 4x + 5 + (7x + 4)(n - 1) \] \[ a_n = 4x + 5 + (7x + 4)n - (7x + 4) \]

Combine like terms:
\[ a_n = (4x - 7x - 4) + (7x + 4)n + 5 \]
\[ a_n = (-3x - 4) + (7x + 4)n + 5 \]

Simplify further:
\[ a_n = (7x + 4)n + (-3x - 4 + 5) \]
\[ a_n = (7x + 4)n + (-3x + 1) \]

Thus, the general term of the sequence is:
\[ \boxed{a_n = (7x + 4)n - 3x + 1} \]

Calculating the 7th Term

With the general formula in hand, we can now find the 7th term \( a_7 \).

Step 1: Substitute \( n = 7 \) into the formula

\[ a_7 = (7x + 4) \times 7 - 3x + 1 \]

Step 2: Expand and Simplify

Compute: \[ a_7 = 7 \times (7x + 4) - 3x + 1 \] \[ a_7 = 49x + 28 - 3x + 1 \]

Combine like terms:
\[ a_7 = (49x - 3x) + (28 + 1) \]
\[ a_7 = 46x + 29 \]

Therefore, the 7th term of the sequence is:
\[ \boxed{a_7 = 46x + 29} \]

Special Cases and Additional Considerations

Depending on the value of \( x \), the sequence's behavior varies. Let's explore some scenarios.

When \( x \) is a specific value

• For example, if \( x = 1 \):

\[ a_7 = 46(1) + 29 = 46 + 29 = 75 \]

• If \( x = 0 \):

\[ a_7 = 46(0) + 29 = 29 \]

• If \( x = -1 \):

\[ a_7 = 46(-1) + 29 = -46 + 29 = -17 \]

Implications of Different \( x \) Values

The sequence's terms depend linearly on \( x \), highlighting the importance of understanding the variable's role. When \( x \) is known, the exact 7th term can be calculated directly; otherwise, the expression remains in terms of \( x \).

Summary and Key Takeaways

• The sequence \( 4x + 5, 11x + 9, 18x + 13, ... \) is arithmetic with a common difference \( d = 7x + 4 \).
• The general term of the sequence is \( a_n = (7x + 4)n - 3x + 1 \).
• The 7th term is \( a_7 = 46x + 29 \).
• The value of \( a_7 \) depends on the variable \( x \), emphasizing the importance of knowing \( x \) for exact calculations.

Practical Applications of Finding Specific Terms in Arithmetic Sequences

Understanding how to find specific terms is useful in various real-world contexts:

• Financial Planning: Calculating future savings or payments based on fixed increments.
• Scheduling: Determining future events occurring at regular intervals.
• Science and Engineering: Modeling phenomena that progress linearly over time or space.

Mastering the process of deriving terms in an arithmetic sequence enhances problem-solving skills and prepares you for more complex mathematical concepts.

Conclusion

Finding the 7th term of the sequence \( 4x + 5, 11x + 9, 18x + 13, ... \) involves recognizing the sequence's arithmetic nature, deriving the general term, and substituting the specific position. The key steps include identifying the common difference, formulating the general term, and evaluating it at \( n=7 \). This systematic approach not only helps in solving this particular problem but also provides a foundation for tackling similar sequence-related questions in mathematics. Remember, understanding the underlying pattern is crucial in working with sequences, and with practice, these calculations become straightforward and intuitive.