Question

In Problems 25-30, find the indicated term in each arithmetic sequence. $$ \text { 100th term of } 2,4,6, \ldots $$

Short answer

The 100th term is 200.

Step-by-step solution

- Identify the First Term

The first term of the arithmetic sequence is given as 2. We will denote the first term by \(a_1\). Therefore, \(a_1 = 2\).

- Determine the Common Difference

The common difference \(d\) can be found by subtracting the first term from the second term. Thus, \(d = 4 - 2 = 2\).

- Use the Formula for the nth Term

To find the nth term of an arithmetic sequence, the formula is: \( a_n = a_1 + (n - 1) \times d \). Here, we need to find the 100th term, so \(n = 100\).

- Substitute the Values into the Formula

Substitute the known values \(a_1 = 2\), \(d = 2\), and \(n = 100\) into the formula: \( a_{100} = 2 + (100 - 1) \times 2 \).

- Simplify the Expression

Simplify the expression step-by-step: \( a_{100} = 2 + 99 \times 2 \) \( a_{100} = 2 + 198 \) \( a_{100} = 200 \).

Key concepts

common difference

In an arithmetic sequence, the 'common difference' is the constant value that separates each term from the next. To find the common difference, you simply subtract the first term from the second term. In the exercise given, the sequence is 2, 4, 6, ...
You can find the common difference, denoted as d, by computing:
\[ d = 4 - 2 = 2 \]
This tells us that each term in the sequence increases by 2. The common difference is crucial because it helps us to identify the pattern and formulate the nth term of the sequence. It's like the 'step' size by which you move from one term to the next in the sequence.

nth term formula

The 'nth term formula' for an arithmetic sequence helps you find the value of any term in the sequence without listing all the terms. The formula to find the nth term (\(a_n\)) is:
\[ a_n = a_1 + (n - 1) \times d \]
Here,

• \(a_n\) is the nth term you're looking for
• \(a_1\) is the first term
• \(n\) is the term number
• \(d\) is the common difference

For our problem, we want to find the 100th term. So, \(a_1 = 2\), \(d = 2\), and \(n = 100\). Plugging these values into the formula, we get:
\[ a_{100} = 2 + (100 - 1) \times 2 \]

sequence identification

Identifying an arithmetic sequence involves recognizing a pattern where each term is obtained by adding a constant. In the provided example, 2, 4, 6, ... , we notice that each number is 2 units greater than the previous one. This constancy confirms that it is an arithmetic sequence. Steps to confirm:

• Check the difference between consecutive terms: \[4 - 2 = 2\]
• Ensure that this difference is consistent throughout: \[6 - 4 = 2\]

The consistency of the common difference indicates that this sequence is arithmetic. Thus, knowing how to identify an arithmetic sequence allows us to apply relevant formulas effectively.

term substitution

Substitution is a straightforward but essential step to find an exact term in an arithmetic sequence. Once we have our formula and known values, substituting these values gives us the desired term. In our example, using the formula for the nth term, the substitution steps are:
1. Our formula: \[a_{100} = 2 + (100 - 1) \times 2\]
2. Calculate inside the brackets: \[100 - 1 = 99\]
3. Multiply by the common difference: \[99 \times 2 = 198\]
4. Add this to the first term: \[2 + 198 = 200\]
Therefore, the 100th term, \(a_{100}\), in the sequence is 200. This step-by-step substitution confirms our understanding and computes the exact term we need.