Problem 49 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{5}=-4, a_{8}=-2.5$$

Step-by-Step Solution

Answer

The general term is \(a_n = 0.5n - 6.5\).

Step 1: Understand the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference," denoted by \(d\). We need to find the general term \(a_n\) for this sequence.

Step 2: Write the General Formula

The general formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \, d\), where \(a_1\) is the first term and \(d\) is the common difference. We need to determine these values or another way to express the general term.

Step 3: Use Given Terms to Find the Common Difference

We use the formula \(a_n = a_1 + (n-1) \, d\) for the given terms. For \(a_5 = -4\), the equation is \(a_1 + 4d = -4\). For \(a_8 = -2.5\), the equation is \(a_1 + 7d = -2.5\). We solve these simultaneously.

Step 4: Subtract Equations to Solve for \(d\)

Subtract the first equation from the second: \((a_1 + 7d) - (a_1 + 4d) = -2.5 - (-4)\). This simplifies to \(3d = 1.5\). Solve for \(d\) by dividing each side by 3: \(d = 0.5\).

Step 5: Substitute \(d\) to Find \(a_1\)

Substitute \(d = 0.5\) back into one of the equations. Using \(a_1 + 4 \cdot 0.5 = -4\) gives \(a_1 + 2 = -4\). Solving for \(a_1\) gives \(a_1 = -6\).

Step 6: Write the General Term \(a_n\)

Now that we know \(a_1 = -6\) and \(d = 0.5\), we can write the general term as \(a_n = a_1 + (n-1) \cdot d = -6 + (n-1) \cdot 0.5\). Simplifying, \(a_n = -6 + 0.5n - 0.5\), so \(a_n = 0.5n - 6.5\).

Key Concepts

Common Difference

The common difference is a crucial aspect of an arithmetic sequence. It represents the consistent interval between any two consecutive terms. In simple terms, it's how much you add (or subtract) each time to move from one term to the next in the sequence. To identify the common difference, denoted as \(d\), you subtract any term from the term that follows it. For example:

• In a sequence like 2, 4, 6, 8, the common difference \(d\) is 2, since 4 - 2 = 2, 6 - 4 = 2, and so on.
• In our exercise, the given terms \(a_5 = -4\) and \(a_8 = -2.5\) were used to find that \(d = 0.5\).

Understanding the common difference allows you to unravel the pattern in the sequence and helps in forming the general term or identifying unknown sequence terms.

General Term Formula

The general term of an arithmetic sequence is like a blueprint formula that lets you calculate any term in the sequence without listing them all. This formula is expressed as:\[a_n = a_1 + (n-1) \, d\]Where:

• \(a_n\) is the \(n\)-th term in the sequence.
• \(a_1\) stands for the first term.
• \(d\) is the common difference.

In our case, we determined the first term \(a_1 = -6\) and the common difference \(d = 0.5\). These values let us write the general term for our sequence: \(a_n = 0.5n - 6.5\). Thus, for any term number \(n\), you can find the sequence value by plugging \(n\) into this formula.

Solving Simultaneous Equations

Simultaneous equations come into play when more than one condition or equation is provided, and you need to solve for common unknowns. In our exercise, we were given:

• \(a_5 = -4\)
• \(a_8 = -2.5\)

Each can be set into the general term form:\[a_1 + 4d = -4\]\[a_1 + 7d = -2.5\]To find the common difference \(d\), we subtract the first equation from the second:\[(a_1 + 7d) - (a_1 + 4d) = -2.5 + 4\]This simplifies to:\[3d = 1.5\]From here, solving gives \(d = 0.5\). This demonstrates how subtracting or equating equations can help find unknown values, which are essential in defining the sequence fully.

Sequence Terms

Understanding sequence terms is fundamental to grasping sequences generally. Each term in the arithmetic sequence is derived from the first term and the common difference. With the general term formula, you can compute successive terms:Given the formula \(a_n = 0.5n - 6.5\) we can extract terms:

• When \(n = 1\), \(a_1 = -6.5 + 0.5 \cdot 1 = -6\)
• When \(n = 2\), \(a_2 = -6.5 + 0.5 \cdot 2 = -5.5\)
• And so forth.

Each term follows logically from its predecessor by the addition of the common difference \(d\). This approach moves us forward in a steady, predictable pattern—perfect for modeling real-world scenarios like financial forecasting or scheduling.