Problem 27 - Precalculus Mathematics for Calculus

Question

Find the nth term of a sequence whose first several terms are given. $$1,4,7,10, \dots$$

Step-by-Step Solution

Answer

The nth term is \( 3n - 2 \).

Step 1: Identify the Pattern

Observe the given sequence: 1, 4, 7, 10, ... Notice that each term increases by 3 from the previous term. This suggests the sequence is arithmetic with a common difference of 3.

Step 2: Arithmetic Sequence Formula

An arithmetic sequence can be described using the formula: \( a_n = a_1 + (n-1)d \) where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number we are trying to find.

Step 3: Assign Known Values

From the sequence, we know \( a_1 = 1 \) and \( d = 3 \). Substitute these into the arithmetic sequence formula: \( a_n = 1 + (n-1) \times 3 \).

Step 4: Simplify the Formula

Simplify the expression: \( a_n = 1 + 3(n-1) = 1 + 3n - 3 = 3n - 2 \). This is the explicit formula for the nth term of the sequence.

Key Concepts

Common Difference

When dealing with arithmetic sequences, a fundamental concept is the **common difference**. Let's explore what it means using our example sequence: 1, 4, 7, 10, ... The term "common difference" refers to the consistent amount by which each term in the sequence increases. We discover this by subtracting a term from the subsequent one. For instance: - From 4 minus 1, we get 3. - From 7 minus 4, we also get 3. - And from 10 minus 7, again, we see 3. Hence, the common difference (d) is 3 in this case. Understanding this value is crucial because it shapes the characteristic linear progression seen in arithmetic sequences. Always remember: the common difference can be a positive number, a negative number, or even zero, leading to increasing, decreasing, or constant sequences, respectively.

Nth Term Formula

Let's dive into understanding the **nth term formula** for arithmetic sequences. This formula helps us find any term in the sequence without listing all the terms. The general formula for finding the nth term is:\[ a_n = a_1 + (n-1)d \]- \(a_n\) is the term you want to find.- \(a_1\) is the first term of the sequence.- \(d\) is the common difference.- \(n\) is the position of the term in the sequence (for which you are trying to find the value). For our sequence (1, 4, 7, 10, ...), we substitute \(a_1 = 1\) and \(d = 3\) into the formula:\[ a_n = 1 + (n-1)3 \]Simplifying the equation gives us:\[ a_n = 3n - 2 \]With this formula, you can determine the value of any term in the sequence, no matter how big the term number \(n\) is. Just plug in the desired \(n\) and calculate.

Sequence Pattern Identification

Unlocking and understanding arithmetic sequences start with **sequence pattern identification**. Observing the sequence pattern is critical to identifying whether the sequence is arithmetic. Let's review our initial sequence: 1, 4, 7, 10, ... By examining this sequence, we notice that each step from one number to the next involves a consistent addition of 3. This repeated addition signals that we are dealing with an arithmetic sequence. Here’s how to identify such a pattern: - **Check for Consistency**: Calculate the difference between consecutive terms to see if it remains the same throughout the sequence. - **Look for Linearity**: The sequence should increase or decrease in a linear manner, i.e., by the same amount every time. Recognizing this pattern enables us to calculate future terms using our formula, reinforcing the connectivity between observation and application. By identifying the pattern, we lay the groundwork to apply the nth term formula effectively.