Problem 42 - 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

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

Step 1: Identify the Formula

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

Step 2: Substitute Known Values

Given that \(a_1 = 1\) and \(d = 4\), substitute these values into the formula: \(a_n = 1 + (n-1) imes 4\). This simplifies to \(a_n = 4n - 3\).

Step 3: Set Up the Equation

We need to determine whether 11,937 is a term of the sequence. Set the nth term equal to 11,937: \(4n - 3 = 11,937\).

Step 4: Solve for n

Solve the equation \(4n - 3 = 11,937\) for \(n\):\[4n - 3 = 11,937\]\[4n = 11,940\]\[n = \frac{11,940}{4}\]\[n = 2,985\]

Step 5: Verify the Solution

Since \(n = 2,985\) is a positive integer, this means 11,937 is indeed a term in the sequence. Hence, it is the 2,985th term.

Key Concepts

nth Term Formula

An arithmetic sequence is a list of numbers where the difference between consecutive terms remains constant. This constant difference is a foundational property of arithmetic sequences. To find a specific term in this sequence, we use the nth term formula:

• \(a_n = a_1 + (n-1) \times d\),

where:

• \(a_n\) is the nth term,
• \(a_1\) is the first term,
• \(d\) is the common difference,
• \(n\) represents the specific term number in the sequence.

This formula is essential to calculate any term from the sequence if we know the first term and the common difference. In our exercise, by applying this formula, we establish the expression \(a_n = 4n - 3\) from the first term and common difference given in the problem.

Common Difference

The common difference in an arithmetic sequence is the consistent interval between terms. It can be determined by subtracting any term from the subsequent term, labeled as \(d\).

• For example, in the given sequence, if the first term \(a_1 = 1\) and this difference \(d = 4\), it implies each term is attained by adding 4 to its previous term.

This property keeps the sequence linear and predictable. Understanding the common difference helps in constructing the sequence and eases the calculation of any term using the nth term formula. If one is found, verifying this within the sequence ensures the accuracy of the calculations performed.

Sequence Term Verification

To confirm a number is part of an arithmetic sequence, one can substitute the value into the nth term formula and solve for \(n\). For the number to be a part of the sequence, \(n\) should be a positive integer, confirming the term is logically placed within the sequence.

• In the exercise, we set the nth term formula equal to 11,937 and solved for \(n\).

This step includes validating whether the calculated \(n\) is feasible by checking it is a positive whole number. In our example, finding \(n = 2,985\) confirms that 11,937 fits rightly within the sequence as the 2,985th term.

Solving Equations

Solving equations is a crucial skill in mathematics, especially when dealing with sequences and series. It involves isolating the variable of interest - in this case, \(n\).

• We took the simplified nth term formula, \(4n - 3 = 11,937\), and solved for \(n\).

Steps include:

• Rearranging the equation: \(4n = 11,940\) by adding 3 to both sides.
• Solving for \(n\) by dividing both sides by 4 resulting in \(n = 2,985\).

Understanding these steps ensures proficiency in determining specific terms in sequences and highlights the relevance of arithmetic operations in forming mathematical arguments and proofs.