Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence. \(a_{n}=24-4 n\)

Step-by-Step Solution

Answer

The first five terms of the sequence are 20, 16, 12, 8, and 4.

Step 1: Understand the Formula

The explicit formula for the arithmetic sequence is given as \(a_{n} = 24 - 4n\). Here, \(a_{n}\) represents the nth term of the sequence.

Step 2: Find the First Term

Substitute \(n = 1\) into the formula to find the first term.\\[a_{1} = 24 - 4(1) = 24 - 4 = 20\]

Step 3: Calculate the Second Term

Substitute \(n = 2\) to find the second term.\\[a_{2} = 24 - 4(2) = 24 - 8 = 16\]

Step 4: Determine the Third Term

Substitute \(n = 3\) into the formula to calculate the third term.\\[a_{3} = 24 - 4(3) = 24 - 12 = 12\]

Step 5: Find the Fourth Term

Substitute \(n = 4\) to get the fourth term.\\[a_{4} = 24 - 4(4) = 24 - 16 = 8\]

Step 6: Calculate the Fifth Term

Finally, substitute \(n = 5\) into the formula to find the fifth term.\\[a_{5} = 24 - 4(5) = 24 - 20 = 4\]

Key Concepts

Explicit Formula

An explicit formula in mathematics provides a straightforward way to find any term in a sequence using its position number. For an arithmetic sequence, the explicit formula shows the relationship between the terms without needing to calculate all the previous terms. In our example, the formula is given by \(a_{n} = 24 - 4n\). This means for any positive integer value of \(n\), you can directly compute the \(n\)-th term by substituting \(n\) into the formula.
The magic of an explicit formula is its ability to bypass long repetitive calculations, providing immediate results for any term you desire.

Nth Term

The \(n\)-th term refers to a specific position within a sequence. In the context of the explicit formula \(a_{n} = 24 - 4n\), this concept helps us identify what each term in the sequence is. The number \(n\) symbolizes the position of the term within the sequence:

• For \(n = 1\), we find the first term.
• For \(n = 2\), the second term.
• And so on, up to any term \(n\).

By changing the value of \(n\), you can compute different terms easily. Simply plug in the desired \(n\) into the formula to get your result.

Algebra Problems

Solving algebra problems often requires the use of formulas and understanding of variables and constants. In the arithmetic sequence \(a_{n} = 24 - 4n\), algebraic skills are applied to deduce terms by substituting numbers for \(n\). This is essential in various math problems, especially in creating simple, logical steps for solving part of larger problems.
Successfully mastering these algebraic techniques helps in:

• Efficient problem-solving.
• Enhancing logical reasoning.
• Encouraging step-by-step thinking.

These skills are practical and transferable to other types of math problems you may encounter.

Sequence Calculation

Sequence calculation involves determining specific terms within a sequence using a particular formula. For our arithmetic sequence of \(a_{n} = 24 - 4n\):

• To find the first term, substitute \(n=1\): \(a_{1} = 24 - 4(1) = 20\).
• For the second term, substitute \(n=2\): \(a_{2} = 24 - 4(2) = 16\).
• The process continues similarly for \(n=3\), \(n=4\), and \(n=5\), resulting in terms: 12, 8, and 4 respectively.

Each substitution shows how sequence calculations rely on consistency and precision, ensuring each step builds correctly on the previous. This method efficiently determines any term desired, leveraging the power of algebra and logical thinking.