Problem 38 - Algebra and Trigonometry

Question

The first term of an arithmetic sequence is \(1,\) and the common difference is \(4 .\) Is \(11,937\) a term of this sequence? If so, which term is it?

Step-by-Step Solution

Answer

Yes, 11,937 is the 2,985th term of the sequence.

Step 1: Understanding the Sequence

In an arithmetic sequence, each term after the first is obtained by adding the common difference to the previous term. Here, the first term is 1, and the common difference is 4.

Step 2: Write the Formula for the n-th Term

The formula for the n-th term of an arithmetic sequence is given by: \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step 3: Substitute Known Values

Substitute the known values into the formula: \( a_n = 1 + (n-1) imes 4 \).

Step 4: Set Up the Equation

To find if 11,937 is a term in the sequence, set \( a_n = 11,937 \) and form the equation: \[ 11,937 = 1 + (n-1) imes 4 \]

Step 5: Solve for n

Re-arrange and solve the equation: - Subtract 1 from both sides: \( 11,936 = (n-1) imes 4 \).- Divide both sides by 4: \( 2,984 = n-1 \).- Add 1 to both sides to solve for \( n \): \( n = 2,985 \).

Step 6: Verify the Result

Substitute \( n = 2,985 \) into the n-th term formula to verify: - \( a_{2985} = 1 + (2985 - 1) imes 4 \).- Calculate: \( 2984 imes 4 = 11,936 \).- Add 1: \( 11,936 + 1 = 11,937 \).Since the calculation is correct, 11,937 is indeed the 2,985th term of the sequence.

Key Concepts

Common Difference

An arithmetic sequence is defined by its initial term and a constant value known as the common difference. This common difference is crucial because it determines how each subsequent term in the sequence progresses. For the given sequence, the first term is 1 and the common difference is 4.

• Each term is obtained by adding 4 to the previous term.
• This results in the sequence: 1, 5, 9, 13, 17, ..., and so on.

This predictable pattern is what makes the arithmetic sequence simple yet powerful. Understanding the common difference is essential, as it plays a pivotal role not only in defining the sequence but also in finding any arbitrary term within it.

n-th Term Formula

The n-th term formula of an arithmetic sequence allows us to calculate the value of any term based on its position in the sequence. The n-th term formula is given by:
\[ a_n = a_1 + (n-1) \times d \]

• \(a_n\) is the n-th term, which you're solving for or verifying.
• \(a_1\) is the first term of the sequence.
• \(n\) is the term number, representing the position of the term in the sequence.
• \(d\) is the common difference, the amount added to each term.

By using this formula, you can directly compute any term without listing all previous terms. This is particularly useful for sequences where the position number is large.

Solving Equations

To determine whether a specific number, like 11,937, is a term in an arithmetic sequence, we employ the n-th term formula to create an equation.
In this situation, we need to figure out if for some positive integer \(n\), the n-th term equals 11,937:
\[ 11,937 = 1 + (n-1) \times 4 \]
Consider the following steps to solve for \(n\):

• Subtract 1 from both sides to simplify the equation: \(11,936 = (n-1) \times 4\).
• Divide both sides by the common difference 4: \(2,984 = n-1\).
• Solve for \(n\) by adding 1: \(n = 2,985\).

Through solving this equation, we determine that 11,937 is the 2,985th term in the sequence. Hence, verifying whether a number belongs to an arithmetic sequence is just a matter of forming and solving a simple equation.

Sequence Verification

Verifying if a specific term belongs to an arithmetic sequence involves re-applying the n-th term formula. It's crucial for ensuring the solution's accuracy. In this instance, after determining \(n = 2,985\), we substitute back into the formula:
\[ a_{2985} = 1 + (2985 - 1) \times 4 \]

• Calculate \(2984 \times 4 = 11,936\).
• Add the initial term \(a_1\): \(11,936 + 1 = 11,937\).

These calculations confirm that 11,937 is indeed the correct term in the sequence, placing no doubt on our earlier algebraic work. Sequence verification is a valuable step to ensure there are no miscalculations.