Problem 16 - Algebra for College Students

Question

$$ 7,10,13,16,19, \ldots \quad 3 n+4 $$

Step-by-Step Solution

Answer

The nth term is \( 3n + 4 \).

Step 1: Identify the Pattern

Examine the sequence: 7, 10, 13, 16, 19,... You can see that you add 3 to each term to get the next term, implying it's an arithmetic sequence with a common difference of 3. The first term is 7.

Step 2: Arithmetic Sequence Formula

The formula for the nth term of an arithmetic sequence is given by: \[ a_n = a + (n-1) imes d \]where \( a \) is the first term and \( d \) is the common difference. In this sequence, \( a = 7 \) and \( d = 3 \).

Step 3: Find the nth Term

Substitute the known values into the arithmetic sequence formula to find the expression for the nth term:\[ a_n = 7 + (n-1) imes 3 \] \[ a_n = 7 + 3n - 3 \] Simplify the expression:\[ a_n = 3n + 4 \]

Step 4: Verification

Check that the derived expression, \( 3n + 4 \), generates the first few terms of the sequence:\- For \( n=1 \), \( 3(1) + 4 = 7 \)\- For \( n=2 \), \( 3(2) + 4 = 10 \)\- For \( n=3 \), \( 3(3) + 4 = 13 \)This matches the given sequence, thus verifying the formula.

Key Concepts

Understanding the Common Difference

An arithmetic sequence is a string of numbers where each term after the first is obtained by adding a constant called the **common difference**. In the sequence 7, 10, 13, 16, 19, we find this difference by calculating how much is added to each term to reach the next one.

• Subtract the first term from the second: 10 - 7 = 3
• Likewise, subtract the second from the third: 13 - 10 = 3
• This confirms the common difference of 3.

Recognizing this constant difference is crucial when working with arithmetic sequences as it describes the linear growth of the sequence. Mastery of this concept helps in predicting further terms and understanding the sequence's linear nature.

Exploring the Sequence Pattern

A sequence pattern is the regular interval or rule used to generate the terms of a sequence. In an arithmetic sequence, the pattern is simple and based on the common difference.
In our given sequence:

• The sequence starts at 7.
• Each subsequent term is formed by adding 3 to the previous term.
• So the pattern is: Start with 7, then continually add 3.

Understanding the pattern helps in quickly identifying the next terms without computation. This knowledge offers insights into the sequence's behavior and confirms the arithmetic nature through its constant incremental pattern.

Applying the nth Term Formula

The nth term formula of an arithmetic sequence provides a direct way to find any term without listing all the previous ones. It's incredibly useful in efficiently handling larger sequences.The formula is: \[ a_n = a + (n-1) \times d \]where

• \( a \) is the first term,
• \( n \) is the term's position in the sequence,
• \( d \) is the common difference.

In our sequence, substituting the known values gives:\[ a_n = 7 + (n-1) \times 3 \]Simplifying this results in:\[ a_n = 3n + 4 \]This formula allows for the calculation of any term easily. For example, to find the 5th term, plug 5 in for \( n \): \( 3(5) + 4 = 19 \). The formula empowers you to explore vast sequences without cumbersome calculations, confirming patterns or making predictions.