Problem 31 - Contemporary Precalculus

Question

The first term \(a_{1}\) and the common difference d of an arithmetic sequence are given. Find the fifth term and the formula for the nth term. $$a_{1}=8, d=.1$$

Step-by-Step Solution

Answer

Also, provide the general formula for the nth term of this sequence. Answer: The 5th term of the arithmetic sequence is 8.4, and the general formula for the nth term is \(a_n = 8 + 0.1(n-1)\).

Step 1: Identify the Given Information

We are given the first term of the arithmetic sequence, \(a_1 = 8\), and the common difference, \(d = 0.1\).

Step 2: Apply the Arithmetic Sequence Formula

We know that the formula for finding the nth term in an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ Now, we'll apply this formula to find the 5th term and the general formula for the nth term.

Step 3: Find the 5th term

To find the 5th term, we plug in the given values for \(a_1\), \(d\), and \(n = 5\): $$a_5 = a_1 + (5-1)d = 8 + (4)(0.1) = 8 + 0.4 = 8.4$$ Thus, the 5th term of the arithmetic sequence is 8.4.

Step 4: Find the Formula for the nth term

To find the general formula for the nth term, we simply replace the given values of \(a_1\) and \(d\) in the arithmetic sequence formula: $$a_n = 8 + (n-1)(0.1)$$ Therefore, the formula for the nth term of the arithmetic sequence is: $$a_n = 8 + 0.1(n-1)$$

Key Concepts

Common Difference in an Arithmetic Sequence

In an arithmetic sequence, each term after the first is created by adding a constant value called the "common difference" to the previous term. This is a key feature of arithmetic sequences that makes them easy to work with. The common difference is consistent throughout the sequence and can be positive, negative, or even zero—resulting in a constant sequence if it's zero.

To calculate the common difference, simply subtract any term in the sequence from the succeeding one. In our exercise, the common difference is given as 0.1. It means to move from one term to the next, you always add 0.1. Understanding the common difference is essential, as it directly affects each term's value in the sequence.

• Given first term (\(a_1 = 8\)).
• Common difference (\(d = 0.1\)).

By knowing these values, you get to define how the sequence grows, making it easier to calculate any term in the series.

nth Term Formula of an Arithmetic Sequence

The nth term formula of an arithmetic sequence allows you to find any term in the sequence without having to list all the preceding terms. The formula is:\[ a_n = a_1 + (n-1)d \]

Here's what each part of the formula represents:

• \(a_n\) is the term you're looking for in the sequence.
• \(a_1\) is the first term, which marks the sequence's starting point.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference, which shows the consistent step between each term.

For our exercise, using this formula, you substitute the given values:\[ a_n = 8 + (n-1) \times 0.1 \]
This gives us a convenient way to find any term of the sequence, such as the 5th term or the 100th term, just by adjusting the value of \(n\). Understanding this formula helps you find any desired term efficiently and is a cornerstone of mastering arithmetic sequences.

5th Term Calculation in an Arithmetic Sequence

Once you have the nth term formula, calculating any specific term, such as the 5th term, becomes straightforward. Here's how you do it step by step:

1. First, identify n's value. For the 5th term, set \(n = 5\).
2. Use the arithmetic sequence formula: \[ a_n = a_1 + (n-1) \times d \]
3. Plug in the known values: \[ a_5 = 8 + (5-1) \times 0.1 \]
4. Simplify the expression:
- Compute inside the parentheses: 5 - 1 = 4.
- Multiply by the common difference: 4 \times 0.1 = 0.4.
- Add to the first term: \[ a_5 = 8 + 0.4 = 8.4 \]
Thus, the 5th term of this arithmetic sequence is 8.4. This consistency in calculation confirms both the formula and the common difference are applied correctly, reinforcing how arithmetic sequences are predictably structured.