Problem 31 - Algebra and Trigonometry

Question

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ -12,-8,-4,0, \dots $$

Step-by-Step Solution

Answer

Common difference is 4, fifth term is 4, 100th term is 384.

Step 1: Identify the Common Difference

In an arithmetic sequence, the common difference is the amount added to each term to get the next term. To find it, subtract the first term from the second term: \[-8 - (-12) = -8 + 12 = 4.\] So, the common difference is 4.

Step 2: Find the Fifth Term

The fifth term of the sequence can be found by adding the common difference to the fourth term (which is 0) once more: \[0 + 4 = 4.\]Thus, the fifth term is 4.

Step 3: Derive the Formula for the n-th Term

The general formula for the n-th term of an arithmetic sequence is given by: \[a_n = a_1 + (n - 1) imes d,\]where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = -12\) and \(d = 4\). Plug these into the formula: \[a_n = -12 + (n - 1) imes 4.\]

Step 4: Calculate the 100th Term

Using the formula derived for the n-th term, replace \(n\) with 100 to find the 100th term:\[a_{100} = -12 + (100 - 1) imes 4.\]Simplify it: \[a_{100} = -12 + 99 imes 4 = -12 + 396 = 384.\]Thus, the 100th term is 384.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key factor that determines the consistency of the pattern between terms. The common difference, denoted as \(d\), is the difference between consecutive terms in the sequence. It's what you add to one term to get to the next.

To calculate the common difference, pick any term in the sequence and subtract its preceding term. For example, if we have the sequence \(-12, -8, -4, 0, \ldots\), the common difference can be calculated by subtracting the first term \(-12\) from the second term \(-8\).

• Calculation: \(-8 \; - (-12) = -8 + 12 = 4\).

Hence, the common difference \(d\) is \(4\). Every subsequent term increases by this value.

nth Term Formula

Understanding the nth term formula is crucial as it provides a powerful tool to find any term in an arithmetic sequence without listing all the terms. The nth term formula for an arithmetic sequence is expressed as:\[ a_n = a_1 + (n - 1) \times d \]

Here, \(a_1\) represents the first term of the sequence, \(n\) is the term number you wish to find, and \(d\) is the common difference.

• For our sequence: \(-12, -8, -4, 0, \ldots\)
• First term \(a_1 = -12\)
• Common difference \(d = 4\)

Substitute these values into the formula to create a generalized expression for the nth term:

• Formula becomes: \[a_n = -12 + (n - 1) \times 4\]

This formula allows you to find any term position \(n\) in the sequence with ease.

Fifth Term Calculation

The fifth term of an arithmetic sequence is found using the nth term formula by substituting \(n\) with 5. From our sequence and derived formula:\[ a_n = -12 + (n - 1) \times 4 \]To find \(a_5\), substitute 5 for \(n\):

• \[ a_5 = -12 + (5 - 1) \times 4 \]
• Continue by calculating: \[ a_5 = -12 + 4 \times 4 \]
• Simplify: \[ a_5 = -12 + 16 = 4 \]

Thus, the fifth term in the sequence is \(4\). This quick calculation highlights how useful the nth term formula can be for finding specific terms.

100th Term Calculation

Finding the 100th term in an arithmetic sequence might seem complex, but the nth term formula simplifies this task significantly. By using the formula,\[ a_n = -12 + (n - 1) \times 4 \]we can substitute \(n\) with 100 to find the 100th term:

• Substitute: \[ a_{100} = -12 + (100 - 1) \times 4 \]
• Simplify the expression: \[ a_{100} = -12 + 99 \times 4 \]
• Carry out the multiplication: \[ a_{100} = -12 + 396 \]
• Finally, add to find the term: \[ a_{100} = 384 \]

The result shows that the 100th term in this arithmetic sequence is \(384\). This approach demonstrates how efficiently the nth term formula can handle large term positions.