Problem 38 - Precalculus: Mathematical for Calculus

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 2985th term of the sequence.

Step 1: Understanding the Arithmetic Sequence Formula

An arithmetic sequence has the general formula for the nth term: \( a_n = a_1 + (n-1) \cdot d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. In this example, \( a_1 = 1 \) and \( d = 4 \).

Step 2: Setting Up the Equation

We need to determine if 11,937 is a term in this sequence. Hence, we set \( a_n = 11937 \) in the formula: \( 11937 = 1 + (n-1) \cdot 4 \).

Step 3: Isolating the Term Number (n)

Rearrange the equation: \( 11937 = 1 + 4(n-1) \) becomes \( 11937 - 1 = 4(n-1) \). So, \( 11936 = 4(n-1) \).

Step 4: Solving for n

Divide both sides by 4 to solve for \( n-1 \): \( 11936 \div 4 = n-1 \). This simplifies to \( 2984 = n-1 \).

Step 5: Finding n

Solve for \( n \) by adding 1 to both sides: \( n = 2984 + 1 \). Hence, \( n = 2985 \).

Step 6: Verification

Check if the solution is correct by plugging \( n = 2985 \) back into the formula: \( a_n = 1 + (2985-1) \cdot 4 \). Calculate \( a_{2985} = 1 + 2984 \cdot 4 = 1 + 11936 = 11937 \). The calculation confirms that 11,937 is indeed the 2985th term.

Key Concepts

Understanding the nth Term Formula

The nth term formula is crucial for identifying any term in an arithmetic sequence. In simple terms, this formula allows you to find the value or position of a specific term in the sequence. It is shown as:

• \( a_n = a_1 + (n-1) \cdot d \)

Here:

• \( a_n \) represents the nth term you wish to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number or position of the term in the sequence.
• \( d \) is the common difference between consecutive terms.

Understanding this formula aids in calculating any term, especially when you have a long sequence. You simply need the first term, the common difference, and the specific position of the term you're interested in finding. This efficient method prevents manual calculations, thereby saving time.

Identifying the Common Difference

The common difference is a key component of an arithmetic sequence. It represents the consistent difference between each term and the next.To find it, you subtract any term from the next term in the sequence:

• \( d = a_2 - a_1 \)

In arithmetic sequences, this common difference is fixed, meaning it remains the same throughout the sequence. In our exercise example, the common difference is given as \(4\). This tells us that each subsequent term increases by \(4\) units from the previous term. Understanding the common difference helps establish the pattern of the sequence, allowing you to predict future terms or even fill in missing ones.By using the common difference, mathematical certainty is added, ensuring no discrepancies occur while finding specific terms.

Sequence Verification Process

Sequence verification involves determining if a particular value is part of the given arithmetic sequence. This requires using the nth term formula, which helps verify if a term belongs to the sequence and identifies its exact position.Here's how you execute the verification:

• Take the given term value and set it equal to the nth term formula: \( a_n = a_1 + (n-1) \cdot d \).
• Solve for \( n \) to find the term's position in the sequence.
• Check your calculated \( n \) by substituting back into the nth term formula to ensure the term value matches the given value.

In our example, we verified that \(11,937\) is indeed a part of the sequence by finding that it corresponds to the 2985th term. This careful verification avoids errors and confirms the integrity of the sequence calculations. By following these steps, any term can be checked effectively within the sequence.