Problem 48 - Precalculus with Limits

Question

In Exercises 47-62, write an expression for the apparent \(n\)th term of the sequence. (Assume that \( n \) begins with 1.) \( 3, 7, 11, 15, 19, \dots \)

Step-by-Step Solution

Answer

Hence, the expression for the apparent \(n\)th term of the sequence is \(4n - 1\).

Step 1: Identify the common difference

The common difference in arithmetic progression is the difference between any two consecutive terms. Here for the sequence \( 3, 7, 11, 15, 19, \dots \), each term is increased by 4. So the common difference, \(d\), is 4.

Step 2: Identify the first term

The first term, also known as the initial term of an arithmetic sequence is the first number in the sequence. Here, for the sequence \( 3, 7, 11, 15, 19, \dots \), the first term, \(a_{1}\), is 3.

Step 3: Formulate the nth term

The general formula for the nth term of an arithmetic sequence is \(a_{n} = a_{1} + (n-1) \cdot d\). Substituting \(a_{1} = 3\) and \(d = 4\) in the equation, we get \(a_{n} = 3 + (n-1) \cdot 4 = 4n - 1\)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is one of the key features that define the sequence. It represents the constant amount added to each term to get to the next term in the sequence. In the example sequence given: \(3, 7, 11, 15, 19, \ldots\), we notice that each number increases by 4.

• To find the common difference, \(d\), you simply subtract the first term from the second term: \(7 - 3 = 4\).
• You can also find it by subtracting any pair of consecutive terms: \(15 - 11 = 4\).

This consistent difference of 4 tells us that the sequence is moving upward evenly by the same amount. Understanding the common difference helps in analyzing how the sequence progresses.

Nth Term

The nth term of an arithmetic sequence is a way to find any term in the sequence without listing all the terms up to that point. This is particularly useful for large sequences.
Knowing the first term and the common difference, we can calculate the nth term using a formula.

• The nth term of an arithmetic sequence is represented by \(a_n\).
• The formula to find \(a_n\) is: \(a_n = a_1 + (n-1) \cdot d\).
• In our given sequence, the first term \(a_1\) is 3, and the common difference \(d\) is 4.

Substituting these values into the formula gives us: \(a_n = 3 + (n-1) \cdot 4\). This allows us to quickly find out the value of any term in the sequence by just plugging the term number \(n\) into the formula.

Sequence Formula

The sequence formula is the mathematical expression used to describe the general form of the sequence, which can be applied to find any term in the sequence. The formula is foundational for understanding the structure of the sequence.
For the given arithmetic sequence, the formula is derived from our knowledge of the first term and the common difference.

• With the first term \(a_1 = 3\) and the common difference \(d = 4\), the formula becomes \(a_n = a_1 + (n-1) \cdot d\).
• Simplifying this: \(a_n = 3 + 4(n-1)\).
• This simplifies further to \(a_n = 4n - 1\).

This sequence formula \(a_n = 4n - 1\) encapsulates the entire sequence in one neat expression. It indicates that for any term number \(n\), you can find the value of that term simply by substituting \(n\) into the formula. This is incredibly useful for quickly calculating terms without extensive computation.