Problem 27 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

Common difference: 4, Fifth term: 4, \(n\)-th term: \(a_n = 4n - 16\), 100th term: 384.

Step 1: Identify the First Term and the Second Term

The first term of the sequence is given as \(-12\) and the second term is \(-8\). The first term is often denoted as \(a_1 = -12\) and \(a_2 = -8\).

Step 2: Calculate the Common Difference

The common difference \(d\) in an arithmetic sequence is found by subtracting the first term from the second term:\[d = a_2 - a_1 = -8 - (-12) = -8 + 12 = 4.\]So, the common difference \(d\) is \(4\).

Step 3: Determine the Fifth Term

Use the common difference to calculate the fifth term. The formula for the \(n\)-th term of an arithmetic sequence is:\[a_n = a_1 + (n-1)d.\]For the fifth term \(a_5\):\[a_5 = a_1 + (5-1)d = -12 + 4 \times 4 = -12 + 16 = 4.\]So, the fifth term is \(4\).

Step 4: Determine the General Formula for the \(n\)-th Term

Using the formula for the \(n\)-th term:\[a_n = a_1 + (n-1)d,\]substitute \(a_1 = -12\) and \(d = 4\) to get the formula:\[a_n = -12 + (n-1) \times 4.\]Simplify it:\[a_n = -12 + 4n - 4 = 4n - 16.\]This is the formula for the \(n\)-th term: \(a_n = 4n - 16\).

Step 5: Determine the 100th Term

Using the general formula \(a_n = 4n - 16\), substitute \(n = 100\) to find the 100th term:\[a_{100} = 4 \times 100 - 16 = 400 - 16 = 384.\]Thus, the 100th term is \(384\).

Key Concepts

Common Difference

An arithmetic sequence is a series of numbers where each term increases or decreases by a constant amount, known as the common difference.
In our example with the sequence \(-12, -8, -4, 0, \ldots\), the first term is \(-12\) and the second term is \(-8\). To find the common difference, we subtract the first term from the second term:

• \[d = a_2 - a_1 = -8 - (-12) = 4\]

This calculation shows that each term in the sequence increases by \(4\). The common difference, \(d\), is crucial because it helps us pinpoint the pattern within the sequence and allows us to calculate later terms using this constant rate of change.

n-th Term Formula

The general formula for finding any term in an arithmetic sequence is essential to work out terms beyond those given. This formula is

• \[a_n = a_1 + (n-1)d\]

where \(a_n\) represents the \(n\)th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
For our sequence, substituting \(a_1 = -12\) and \(d = 4\) gives us:

• \[a_n = -12 + (n-1) \times 4\]

Simplifying further, the formula becomes:

• \[a_n = 4n - 16\]

This formula lets you calculate the value of any term in the sequence without having to list every term before it.

100th Term Calculation

To find a specific term far along in an arithmetic sequence, such as the 100th term, we can plug into the n-th term formula from the previous section. This eliminates the need to calculate each term one by one.
We use the formula:

• \[a_n = 4n - 16\]

Plugging \(n = 100\) into this formula gives us:

• \[a_{100} = 4 \times 100 - 16 = 384\]

Thus, the 100th term in the sequence is \(384\). This approach is quick and efficient, allowing for easy calculation of any term in the sequence.