Problem 48 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=-2, a_{5}=8$$

Step-by-Step Solution

Answer

The general term is \( a_{n} = 2.5n - 4.5 \).

Step 1: Understand what an arithmetic sequence is

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant called the common difference ( ).

Step 2: Use the formula for the general term

The general term of an arithmetic sequence can be given by the formula: \( a_{n} = a_{1} + (n-1)d \).

Step 3: Determine the common difference

We know that \( a_{1} = -2 \) and \( a_{5} = 8 \). Use the formula for the nth term for \( a_{5} \): \( a_{5} = a_{1} + 4d = 8 \).Substitute \( a_{1} = -2 \): \( -2 + 4d = 8 \).Solve for \( d \): \( 4d = 10 \) \( d = \frac{10}{4} \) \( d = 2.5 \).

Step 4: Write out the general term

Now that we have \( d = 2.5 \), substitute \( a_{1} \) and \( d \) back into the formula for the general term:\[ a_{n} = -2 + (n-1)(2.5) \].

Step 5: Simplify the formula

Expand and simplify the expression for \( a_{n} \):\( a_{n} = -2 + 2.5n - 2.5 \)\( a_{n} = 2.5n - 4.5 \).

Key Concepts

Common Difference

Understanding the common difference is crucial when dealing with an arithmetic sequence. In an arithmetic sequence, the common difference, denoted as \(d\), is the amount by which successive terms in the sequence increase (or decrease) as you move from one term to the next.
For example, in the sequence \(2, 4, 6, 8\), the common difference is \(2\), because you add 2 to each term to get the next one.

• Positive Common Difference: This means the sequence increases.
• Negative Common Difference: Leads to a decreasing sequence.
• Zero Common Difference: The same term repeats indefinitely.

In the original exercise, finding the common difference helped us bridge from the first term to the fifth term, leading to \(d = 2.5\). This illustrated that despite the arithmetic beginning with \(-2\), the sequence rose by \(2.5\) units per step.

General Term Formula

The general term formula is your map to the infinite landscape of an arithmetic sequence. It allows you to find the value of any term in the sequence without listing all the previous terms.
The formula is given by: \[ a_{n} = a_{1} + (n-1)d \]
Where:

• \(a_{n}\): The term you want to find.
• \(a_{1}\): The first term in the sequence.
• \(n\): The term number.
• \(d\): The common difference.

This formula helps decode the position of any desired term in the sequence. For instance, you can determine the tenth term directly from the formula without the tedium of computing terms 2 through 9 first. In the provided problem, after calculating \(d\), substituting \(a_1 = -2\) and \(d = 2.5\) into the formula enabled us to represent the sequence by \(a_{n} = 2.5n - 4.5\).

Algebraic Problem Solving

Using algebra to solve problems related to arithmetic sequences enhances our problem-solving toolkit. By setting up equations based on known information (like the exercise's first and fifth terms), we can systematically find unknowns.
The process often involves:

• Identifying known values (e.g., \(a_{1} = -2\) and \(a_{5} = 8\)).
• Applying them to the general term formula \(a_{5} = a_{1} + 4d = 8\).
• Simplifying the equation to find the unknown common difference \(d\).

The problem essentially becomes a linear equation, solvable using basic algebra. Solving for \(d\) allows us to express the arithmetic sequence comprehensively. Algebra transforms these sequences from simple sets of numbers into understandable mathematical narratives, enriching our comprehension and enabling predictions.