Problem 47 - 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.) \( 1, 4, 7, 10, 13, \dots \)

Step-by-Step Solution

Answer

The apparent nth term of the sequence 1, 4, 7, 10, 13, .... is \( a_n = 3n - 2 \)

Step 1: Identify Sequence Type

The numbers provided form an arithmetic sequence, where each subsequent number is obtained by adding a constant difference. In this case, the difference is 3

Step 2: Identify First Term

The first term (n=1) of the sequence given is \( a_1=1 \)

Step 3: Generate the nth term

The nth term of an arithmetic sequence can be obtained using the formula \( a_n = a_1 + (n - 1) \cdot d \) where \( a_1 \) is the first term, \( d \) is the common difference and \( n \) is the term number. Substituting the identified values into this formula, the nth term can be denoted as \( a_n = 1 + (n - 1) \cdot 3 \)

Step 4: Simplify the nth term

Simplify the formula for \( a_n \) to get a more understandable form. If we multiply out we get \( a_n = 1 + 3n - 3 \). This simplifies further to \( a_n = 3n - 2 \)

Key Concepts

nth term of a sequence

Understanding the "nth term" of a sequence is vital for analyzing and predicting any number in a sequence without manually calculating each prior term. In the context of arithmetic sequences, which are sequences of numbers with a constant difference between consecutive terms, the formula for the nth term is an extremely useful tool.
The formula for finding the nth term of an arithmetic sequence is:

• \( a_n = a_1 + (n - 1) \cdot d \)

Here:

• \( a_n \) is the nth term you want to find,
• \( a_1 \) is the first term in the sequence,
• \( d \) is the common difference,
• and \( n \) is the position of the term in the sequence.

The formula allows you to plug in the values you know and solve for the term you need. In the exercise you provided, the sequence was: 1, 4, 7, 10, 13, ... The nth term can be determined by substituting into the formula: \( a_n = 1 + (n - 1) \cdot 3 \), leading to the simplified form of \( a_n = 3n - 2 \). This allows us to find any term in the sequence by simply substituting the position of the term for \( n \).

common difference

The "common difference" is the backbone of an arithmetic sequence. It's what distinguishes arithmetic sequences from other types of numerical sequences. Simply put, the common difference is the consistent gap between each term in the sequence.

To determine this, look at any pair of consecutive terms in the sequence. In our sequence example: 1, 4, 7, 10, 13,... we see that each term is obtained by adding 3 to the previous term. Therefore, the common difference, \( d \), is 3.

• It is calculated as \( d = a_{n+1} - a_n \), where \( a_{n+1} \) and \( a_n \) are consecutive terms.

This common difference plays a significant role not only in generating the sequence but also in the formula for the nth term. It's multiplied by \((n-1)\) when determining the nth term. Understanding the common difference is crucial as it ensures you correctly build and understand the structure of the sequence.

first term of a sequence

The "first term" of a sequence is where it all begins. It acts as the starting point, setting the entire sequence in motion. For arithmetic sequences, identifying the first term is crucial because it serves as the basis for all subsequent terms.

In the given sequence example: 1, 4, 7, 10, 13,... the first term, \( a_1 \), is 1. It represents the value of the sequence when \( n = 1 \). Remember, the first term is not just a number; it is part of the key foundational information needed to apply the nth term formula successfully. Without it, you cannot reliably calculate the future terms in the sequence.

• Formally, this can be denoted as \( a_1 \).

Having a clear understanding of what the first term represents ensures that when you employ the nth term formula, the sequence generated will accurately reflect the actual values of the given sequence. Thus, when working with arithmetic sequences, always start by identifying and clearly noting the first term.