Problem 43 - Precalculus: Mathematics for Calculus

Question

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$2,2+s, 2+2 s, 2+3 s, \dots$$

Step-by-Step Solution

Answer

Common difference: \(s\); Fifth term: \(2 + 4s\); \(n\)th term: \(2 + (n-1)s\); 100th term: \(2 + 99s\).

Step 1: Identify the First Term

The first element of the sequence is given as 2. So, the first term, commonly denoted as \(a\), is \(a = 2\).

Step 2: Determine the Common Difference

The arithmetic sequence has consecutive terms 2, \(2 + s\), \(2 + 2s\), \(2 + 3s\). The common difference \(d\) is the difference between any two consecutive terms. Here, \(d = (2+s) - 2 = s\).

Step 3: Find the Fifth Term

The general formula for the \(n\)th term \(T_n\) in an arithmetic sequence is \(T_n = a + (n-1) \cdot d\). To find the fifth term, substitute \(n = 5\), \(a = 2\), and \(d = s\) into the formula:\[T_5 = 2 + (5-1) \cdot s = 2 + 4s.\]

Step 4: Derive the General Expression for the \(n\)th Term

Using the formula for the \(n\)th term, \(T_n = a + (n-1) \cdot d\), substitute \(a = 2\) and \(d = s\):\[T_n = 2 + (n-1) \cdot s = 2 + (n-1)s.\]

Step 5: Compute the 100th Term

We use the general term formula: \(T_n = 2 + (n-1) \cdot s\). Substitute \(n = 100\), \(a = 2\), and \(d = s\):\[T_{100} = 2 + (100-1) \cdot s = 2 + 99s.\]

Key Concepts

Common Difference

An arithmetic sequence progresses by adding a fixed amount to the previous term to form the next one. This fixed amount is known as the "common difference." To identify the common difference in any sequence:

• Look at two consecutive terms.
• Subtract the first term from the second.

For the given sequence, the terms are: 2, \(2 + s\), \(2 + 2s\), \(2 + 3s\), ....

The common difference, denoted as \(d\), can be calculated by subtracting the first term from the second term:
\[d = (2 + s) - 2 = s\]

So, every term in this sequence increases by \(s\), which is the common difference. Understanding this helps simplify finding any term in the sequence.

nth Term

The formula for finding any term in an arithmetic sequence is based on the position of that term, known as the "nth term." The nth term, \(T_n\), can be calculated using the formula:
\[T_n = a + (n-1) \, \cdot \, d\]
Where:

• \(a\) is the first term of the sequence.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference.

In our example, we determined \(a\) as 2 and \(d\) as \(s\). Plugging these into the formula for the nth term:
\[T_n = 2 + (n-1) \, \cdot \, s\]

This expression means that any term in this sequence is the first term 2, plus the product of the common difference \(s\) and one less than the term position \(n\).
The beauty of the nth term formula is its ability to find any term without listing all previous terms.

100th Term

Finding a specific term in a sequence, like the 100th term, is about using the generalized formula for the nth term. Once we know \(a\) and \(d\), we can find the 100th term in the sequence without extensive computation. Given that \(a = 2\) and \(d = s\), we apply:
\[T_{100} = 2 + (100-1) \, \cdot \, s = 2 + 99s\]

This calculation simplifies finding the 100th term to a straightforward substitution, yielding \(2 + 99s\).
Using the nth term formula saves time and effort in analyzing long arithmetic sequences, where calculating each term individually would be impractical. It ensures you can tackle any position with ease, even the 100th term.