Problem 24 - College Algebra

Question

For the following exercises, find the specified term given two terms from an arithmetic sequence. $$ a_{1}=33 \text { and } a_{7}=-15 . \text { Find } a_{4} $$

Step-by-Step Solution

Answer

The fourth term \(a_4\) is 9.

Step 1: Understanding the Problem

We are given two terms from an arithmetic sequence: the first term \(a_1=33\) and the seventh term \(a_7=-15\). We need to find the fourth term \(a_4\).

Step 2: Recall the Formula for an Arithmetic Sequence

An arithmetic sequence follows the formula: \(a_n = a_1 + (n-1) imes d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Step 3: Find the Common Difference

Using the formula for the seventh term \(a_7 = a_1 + 6d\), substitute the known values:\[a_7 = -15 = 33 + 6d\]Solve for \(d\):\[6d = -15 - 33 \implies 6d = -48 \implies d = -8\]

Step 4: Use the Common Difference to Find the Fourth Term

Now that we have \(d=-8\), use the arithmetic sequence formula to find \(a_4\):\[a_4 = a_1 + 3d = 33 + 3(-8)\]Calculate \(a_4\): \[a_4 = 33 - 24 = 9\]

Step 5: Verify the Calculation

Ensure the calculations are correct by verifying another term. Check with \(a_7 = -15\) using \(a_1 = 33\) and \(d = -8\):\[a_7 = a_1 + 6d = 33 + 6(-8) = 33 - 48 = -15\]The sequence is consistent.

Key Concepts

First Term

In an arithmetic sequence, the first term, often denoted as \( a_1 \), acts as the starting point from which the rest of the sequence is generated. It gives us the initial value of the sequence and is crucial for calculating subsequent terms. For any sequence, \( a_1 \) is the value at which the sequence begins before any additional adjustments by way of the common difference.Understanding \( a_1 \) is essential because:

• It grounds the sequence and gives it a starting point.
• All terms in the sequence build on this first term using the common difference.

For example, in the given exercise, the first term \( a_1 \) is 33. Without this first term, it would be impossible to construct the sequence or find other terms like the fourth or seventh term accurately.

Common Difference

The common difference in an arithmetic sequence, often symbolized by \( d \), signifies the uniform change between consecutive terms. It remains constant throughout the sequence. This constancy is what defines an arithmetic sequence, setting it apart from other types of sequences where the change might not be consistent.Calculating the common difference involves:

• Subtraction between any two consecutive terms.
• Using the equation \( a_n = a_1 + (n-1) \times d \) with known terms to solve for \( d \).

In our exercise scenario, given \( a_1 = 33 \) and \( a_7 = -15 \), we calculated \( d = -8 \). This common difference of \(-8\) tells us that each term decreases by 8. We use this information to find other terms like the fourth term of the sequence.

Nth Term

The \( n \)th term of an arithmetic sequence, represented as \( a_n \), is found using the formula \( a_n = a_1 + (n-1) \times d \). This expression allows us to calculate any term in the sequence quickly, once we know the first term \( a_1 \) and the common difference \( d \). By plugging in the appropriate values for \( n \), you can find the specific term you’re interested in.Key points about calculating the \( n \)th term:

• Determine what \( n \) is, which identifies which term you’re calculating.
• Use the formula to substitute \( a_1 \), \( d \), and \( n \) to solve for \( a_n \).

For instance, in the problem, to find \( a_4 \), we used the formula with \( a_1 = 33 \), \( d = -8 \), and \( n = 4 \), resulting in \( a_4 = 9 \). This formula is essential for navigating arithmetic sequences as it directly links the position of the term with its value.