Problem 35 - 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 Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant. The sequence given is \( 2, 2+s, 2+2s, 2+3s, \ldots \). By subtracting the first term from the second term, we find the common difference to be \( s \). Thus, the common difference \( d \) is \( s \).

Step 2: Determine the Fifth Term

The fifth term of the sequence can be found by adding four times the common difference to the first term. The formula for the \( n \)th term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \). For the fifth term \( a_5 \), we have:\[a_5 = 2 + 4s\]So, the fifth term is \( 2 + 4s \).

Step 3: Derive the General Formula for the \( n \)th Term

Using the formula for the \( n \)th term of an arithmetic sequence, \( a_n = a_1 + (n-1)d \), where \( a_1 = 2 \) and \( d = s \), we find the \( n \)th term to be:\[a_n = 2 + (n-1)s\]This is the general formula for the \( n \)th term of the sequence.

Step 4: Calculate the 100th Term

Substitute \( n = 100 \) into the general formula \( a_n = 2 + (n-1)s \) to find the 100th term:\[a_{100} = 2 + 99s\]Hence, the 100th term is \( 2 + 99s \).

Key Concepts

Common Difference

In an arithmetic sequence, the Common Difference refers to the constant value you add to each term to get to the next one. Imagine you are climbing a staircase where each step is equally spaced. That spacing is like the common difference in your sequence. To find this difference in any sequence, simply subtract the first term from the second term.

For example, in the sequence given, namely: \( 2, 2+s, 2+2s, 2+3s, \ldots \), start with the first two terms. Subtract the first term from the second term:

• Second term: \( 2+s \)
• First term: \( 2 \)
• Common Difference, \( s = (2+s) - 2 \)

By doing this simple subtraction, we determine the common difference to be \( s \). This common difference helps us understand how the sequence progresses as we move from one number to the next.

Nth Term Formula

The Nth Term Formula is your key to finding any term in your arithmetic sequence. This formula provides a specific way to calculate any term in the order without having to list out all the terms one by one. Imagine having a magic formula that tells you the number at any step of the staircase right away.

The general form of the Nth Term Formula is given by:

• \( a_n = a_1 + (n-1)d \)

Here:

• \( a_n \) represents the term you're trying to find
• \( a_1 \) is the first term of the sequence
• \( n \) is the position of the term within the sequence
• \( d \) stands for the common difference

Applying this to our exercise, the first term \( a_1 \) is \( 2 \), and the common difference \( d \) is \( s \). Thus, the formula for any term, \( a_n \), becomes \[ a_n = 2 + (n-1)s \] which allows you to calculate any term just by plugging in the value of \( n \). This equation is a powerful tool for exploring the sequence without building it step by step.

General Term

The General Term in an arithmetic sequence is, essentially, the representation of the pattern of the sequence. It tells us what the sequence is doing in a simple and efficient way. When you know the sequence follows a certain pattern, this term keeps you at the helm by showing a straightforward expression that can model it.

To find the general term formula, you plug in the details from the sequence into the Nth Term Formula. As previously discussed, the formula is:

• \( a_n = a_1 + (n-1)d \)

By placing the values as follows:

• First term \( a_1 = 2 \)
• Common difference \( d = s \)

You derive the expression \[ a_n = 2 + (n-1)s \]. This formula now represents what's going on in the sequence. It elucidates how you go from the first term to the \( n \)th term. This is why it's termed "general"— it serves for any term within the sequence by simply changing \( n \). Replace \( n \) with any number to find the corresponding term, like the 100th term shown in our solution!