Problem 17 - Algebra for College Students

Question

$$ 2,-1,-4,-7,-10, \ldots--3 n+5 $$

Step-by-Step Solution

Answer

The expression for the n-th term is \(-3n + 5\).

Step 1: Identify the Pattern

Observe the given sequence: \(2, -1, -4, -7, -10, \ldots\). Notice that each term is obtained by subtracting 3 from the previous term, indicating a consistent decrease. This is an arithmetic sequence with a common difference \(d = -3\).

Step 2: Determine the First Term and Common Difference

Identify the first term of the sequence, \(a_1 = 2\), and the common difference \(d = -3\), as established from the pattern.

Step 3: Use the Arithmetic Sequence Formula

The general formula for the \(n\)-th term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) imes d\). Plug in the known values: \(a_n = 2 + (n-1)(-3)\).

Step 4: Simplify to Find the Expression for the \(n\)-th Term

Simplify the expression: \[a_n = 2 + (-3)(n-1) = 2 - 3n + 3 = -3n + 5\]. Thus, the expression for the \(n\)-th term is \(-3n + 5\).

Step 5: Verify the Expression with Specific Values

Check the expression by substituting specific values of \(n\):- For \(n=1\), \(-3(1) + 5 = 2\)- For \(n=2\), \(-3(2) + 5 = -1\)- For \(n=3\), \(-3(3) + 5 = -4\)The values match the given sequence, confirming the expression is correct.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the constant amount each term is subtracted or added to get the next term. Identifying this difference is key to understanding how the sequence progresses. In our given sequence \(2, -1, -4, -7, -10, \ldots\), each subsequent number is obtained by subtracting 3 from the previous one.
This means the common difference, denoted by \(d\), is \(-3\). This negative difference indicates that the sequence decreases with every term.
Understanding the common difference helps to determine the direction and rate of the sequence, giving insight into the overall pattern and behavior.

Sequence Pattern

The sequence pattern in an arithmetic sequence refers to the predictable way each term is generated from the previous one by using the common difference. For the sequence \(2, -1, -4, -7, -10, \ldots\), there's a visible pattern when you continue subtracting 3 from each term.
Here's how it works:

• Start with the initial term, which is 2.
• Subtract 3 to get the next term (-1).
• Continue this subtraction to get each subsequent term (-4, -7, -10 etc.).

Recognizing the sequence pattern not only helps in predicting upcoming terms but also aids in acknowledging the sequence's consistency. Even without calculation, you can quickly determine that the next few terms would be \(-13, -16, -19,\) and so forth.

Nth Term Formula

To find any term in an arithmetic sequence without having to list all terms, we use the formula for the \(n\)-th term. This is particularly useful for finding distant terms directly. In our example, the formula for the \(n\)-th term \(a_n\) of the sequence is given by:
\[a_n = a_1 + (n-1) \times d\]
where:

• \(a_1\) is the first term of the sequence (which is 2 in our case).
• \(d\) is the common difference (\(-3\)).
• \(n\) is the position of the term in the sequence.

Plugging in the values, the formula becomes:
\[a_n = 2 + (n-1) \times (-3)\]
Simplification leads to:
\[a_n = -3n + 5\]
This compact formula allows you to effortlessly calculate any term in the sequence by simply inserting the term number \(n\), confirming the effectiveness and power of using arithmetic formulas.