Problem 54 - Essential Calculus

Question

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(3,7,11,15, \ldots\)

Step-by-Step Solution

Answer

The expression for the n-th term of the sequence is \(a_n = -1 + 4n\).

Step 1: Find the common difference

The common difference can be found by subtracting any term from the previous term. Here, subtracting the first term (3) from the second term (7) gives us a common difference (d) of 4.

Step 2: Formulate the sequence expression

We can formulate the general expression for arithmetic sequence, which is \(a_n = a_1 + (n-1)*d\). Here, \(a_1\) is the first term of the sequence, and \(d\) is the common difference. Substituting these values into the formula gives us the n-th term expression: \(a_n = 3 + (n-1)*4.\)

Step 3: Simplify the sequence expression

We can simplify this expression further by distributing the 4 in the equation. The simplified expression will look like this: \(a_n = -1 + 4n\).

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the **common difference** is the consistent interval between consecutive terms. It's a crucial part of defining how the sequence progresses. For example, given the sequence: 3, 7, 11, 15, ..., we can find this interval by subtracting any term from the one that follows it. - Subtracting the first term (3) from the second term (7), we get: \[ d = 7 - 3 = 4 \] - This tells us that each term is 4 more than the previous one. It's important to check this pattern holds true across the sequence to confirm that it is, indeed, arithmetic. So subtracting the third term from the second also yields 4 (11 - 7).
The common difference is like the rulebook of our sequence! Without it, the sequence would not follow a steady pattern.

Exploring the n-th Term Formula

Once the common difference is determined, we can derive the equation that defines any term in the arithmetic sequence. This is called the **n-th term formula**. It allows us to find any term in the sequence without listing all previous terms. The formula is given by: \[ a_n = a_1 + (n-1)\times d \] In this formula: - **\(a_n\)** is the term we want to find.

- **\(a_1\)** is the first term of the sequence, which in our sequence is 3. - **\(d\)** is the common difference, which we've found to be 4.

- **\(n\)** is the position of the term in the sequence. Plugging these values into the formula gives us our specific n-th term expression: \[ a_n = 3 + (n-1)\times 4 \]This equation will help us pinpoint any term we want, making it a powerful tool for understanding and predicting the sequence.

Formulating and Simplifying the Sequence Expression

With our n-th term formula in hand, the next step is to simplify it for ease of use. This gives us the **sequence expression**, a practical version of our formula that can be used to quickly find terms. Starting from: \[ a_n = 3 + (n-1)\times 4 \]we simplify by distributing the 4 across the \((n-1)\) which results in: \[ a_n = 3 + 4n - 4 \] This can be combined further to: \[ a_n = 4n - 1 \] Now, this expression is a simplified representation that calculates any term based on its position \(n\), easily representing the sequence's pattern. This expression is more straightforward and highlights the linear nature of arithmetic sequences.