Problem 17 - Beginning and Intermediate Algebra

Question

Write the first five terms of the arithmetic sequence with general term \(a_{n}\). $$a_{n}=6 n+7$$

Step-by-Step Solution

Answer

The first five terms of the arithmetic sequence with the given general term \(a_n = 6n + 7\) are 13, 19, 25, 31, and 37.

Step 1: Compute the first term, \(a_1\)

To find the first term of the arithmetic sequence, we'll plug in n = 1 into the general term formula: $$a_1 = 6(1) + 7$$ $$a_1 = 6 + 7$$ $$a_1 = 13$$

Step 2: Compute the second term, \(a_2\)

To find the second term of the arithmetic sequence, we'll plug in n = 2 into the general term formula: $$a_2 = 6(2) + 7$$ $$a_2 = 12 + 7$$ $$a_2 = 19$$

Step 3: Compute the third term, \(a_3\)

To find the third term of the arithmetic sequence, we'll plug in n = 3 into the general term formula: $$a_3 = 6(3) + 7$$ $$a_3 = 18 + 7$$ $$a_3 = 25$$

Step 4: Compute the fourth term, \(a_4\)

To find the fourth term of the arithmetic sequence, we'll plug in n = 4 into the general term formula: $$a_4 = 6(4) + 7$$ $$a_4 = 24 + 7$$ $$a_4 = 31$$

Step 5: Compute the fifth term, \(a_5\)

To find the fifth term of the arithmetic sequence, we'll plug in n = 5 into the general term formula: $$a_5 = 6(5) + 7$$ $$a_5 = 30 + 7$$ $$a_5 = 37$$ The first five terms of the arithmetic sequence with the given general term are 13, 19, 25, 31, and 37.

Key Concepts

General Term Formula

The general term formula of an arithmetic sequence is a powerful tool to find any term in the sequence. For this specific arithmetic sequence, the general term is given by \(a_n = 6n + 7\). This formula lets us calculate the value of any term in the sequence simply by plugging in the value of \(n\).
Here, the constant "6" represents the common difference, which tells us how much each term differs from the previous. The "7" is the first term adjusted when \(n\) equals zero in a sense.

• Use the general term formula to identify any term without listing out the entire sequence.
• "\(n\)" is the term number or position in the sequence.

Understanding how to use the general term formula is essential for dealing with sequences efficiently.

First Five Terms

To find the first five terms of an arithmetic sequence using the general term formula, we substitute \(n = 1\) through \(n = 5\) into the equation \(a_n = 6n + 7\). This gives each specific term:

• \(a_1 = 6(1) + 7 = 13\)
• \(a_2 = 6(2) + 7 = 19\)
• \(a_3 = 6(3) + 7 = 25\)
• \(a_4 = 6(4) + 7 = 31\)
• \(a_5 = 6(5) + 7 = 37\)

Each term is calculated by plugging the position of the term into the sequence formula. Once you have the formula, finding terms becomes straightforward.
This method is efficient and useful if you need several terms quickly without manually calculating each step.

Beginning Algebra

Beginning algebra involves working with formulas and solving equations with variables like \(n\). Understanding sequences and their general term formulas is part of this. In simple terms, an arithmetic sequence is a list of numbers increasing by the same amount each step.

• Algebra helps to express these consistent changes through formulas like \(a_n = 6n + 7\).
• By identifying the pattern (addition of 6), we set a logical structure to find future terms.
• Using substitution to solve for terms is a foundational algebraic skill.

Practicing these calculations helps build a robust understanding of algebraic expressions and sequence patterns.

Sequence Terms Calculation

Sequence terms calculation can be simplified using the arithmetic sequence formula. Knowing the general term formula \(a_n = 6n + 7\), you can easily derive any term within this sequence.
The key steps involve identifying the position of the term (\(n\)) and substituting it into the formula. For example, for \(n = 4\), plug it in as follows:
\(a_4 = 6(4) + 7 = 31\).

• Select \(n\) based on which term you need.
• Use the formula to compute the term directly.

This reduces errors and provides immediate results compared to extended manual calculations, especially helpful for large sequences. Understanding these concepts speeds up sequence term calculations effectively.