Problem 31 - 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. $$2,2+s, 2+2 s, 2+3 s, \dots$$

Step-by-Step Solution

Answer

Common difference is \(s\), fifth term is \(2 + 4s\), \(n\)-th term is \(2 + (n-1)s\), 100th term is \(2 + 99s\).

Step 1: Identify the pattern

Observe the first few terms of the sequence: 2, \(2+s\), \(2+2s\), \(2+3s\). It seems like each term increases by the same increment, called the common difference, which is the coefficient of \(s\) in each term after the first.

Step 2: Calculate the common difference

The common difference \(d\) in an arithmetic sequence is found by subtracting any term from the subsequent term. Let's subtract the first term from the second term: \((2+s) - 2 = s\). Thus, the common difference \(d = s\).

Step 3: Determine the fifth term

To find the fifth term, use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. For the fifth term: \(a_5 = 2 + (5-1)s = 2 + 4s\). Thus, the fifth term is \(2 + 4s\).

Step 4: Derive a formula for the nth term

Using the same formula for the \(n\)-th term, \(a_n = a_1 + (n-1)d\), substitute \(a_1 = 2\) and \(d = s\) to get: \(a_n = 2 + (n-1)s\). This formula gives any term in the sequence based on its position \(n\).

Step 5: Find the 100th term

Apply the formula of the \(n\)-th term to find the 100th term: \(a_{100} = 2 + (100-1)s = 2 + 99s\). Therefore, the 100th term is \(2 + 99s\).

Key Concepts

Common Difference

In an arithmetic sequence, the "common difference" is the amount each term increases by from the previous one. It is a constant difference, meaning it stays the same throughout the sequence.
To find this common difference, you subtract any term from the next term in the sequence.
In our example sequence, the terms are 2, \(2+s\), \(2+2s\), and \(2+3s\). By subtracting the first term from the second term, we get:

• \((2+s) - 2 = s\)

This means the common difference is \(s\).
Having a common difference allows us to predict what each subsequent term in the sequence will be, showcasing the uniform growth of the arithmetic sequence.

Nth Term Formula

The "nth term formula" of an arithmetic sequence is used to find any term in the sequence without having to list all the terms leading up to it. This formula is written as:

• \(a_n = a_1 + (n-1)d\)

where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
In our specific sequence, the first term \(a_1 = 2\), and the common difference \(d = s\).
Thus, the formula becomes:
\(a_n = 2 + (n-1)s\)
This formula helps you determine any term's value based on its position \(n\), making it a powerful tool for arithmetic sequences.

Arithmetic Sequence Pattern

The "arithmetic sequence pattern" is characterized by numbers in a sequence that increases or decreases by the same constant (the common difference) every step.
The pattern is regular and predictable, which makes arithmetic sequences easy to work with and analyze.
In the example sequence: 2, \(2+s\), \(2+2s\), \(2+3s\), and so on, we can see that each term grows by the common difference \(s\). The beauty of this pattern is its consistency.

• This sequence will always follow the same step to increase the value, allowing us to use the nth term formula effectively.
• Arithmetic sequences allow us to understand, predict, and calculate each term's value if we know the first term and the common difference.

Recognizing this pattern is key to manipulating arithmetic sequences effectively and understanding their behavior.