Problem 5 - Algebra 2

Question

Find the first five terms of each arithmetic sequence described. \(a_{1}=\frac{1}{2}, d=\frac{1}{4}\)

Step-by-Step Solution

Answer

The terms are \( \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} \).

Step 1: Identify the General Formula

The general formula for the n-th term in an arithmetic sequence is given by \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step 2: Calculate the First Term

The first term \( a_1 \) is given as \( \frac{1}{2} \). Thus, the first term of the sequence is \( \frac{1}{2} \).

Step 3: Compute the Second Term

Using the formula \( a_2 = a_1 + d \), substitute \( a_1 = \frac{1}{2} \) and \( d = \frac{1}{4} \) to get \( a_2 = \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \).

Step 4: Calculate the Third Term

Using \( a_3 = a_2 + d \), substitute \( a_2 = \frac{3}{4} \) and \( d = \frac{1}{4} \) to find \( a_3 = \frac{3}{4} + \frac{1}{4} = 1 \).

Step 5: Find the Fourth Term

Next, \( a_4 = a_3 + d \). With \( a_3 = 1 \) and \( d = \frac{1}{4} \), calculate \( a_4 = 1 + \frac{1}{4} = \frac{5}{4} \).

Step 6: Determine the Fifth Term

Finally, \( a_5 = a_4 + d \). Substitute \( a_4 = \frac{5}{4} \) and \( d = \frac{1}{4} \) to get \( a_5 = \frac{5}{4} + \frac{1}{4} = \frac{6}{4} = \frac{3}{2} \).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference, denoted by \( d \), is a fundamental element that gives the sequence its distinct progression. It's the constant interval between successive terms.
For example, if you start with a term \( a_1 = \frac{1}{2} \) and add \( d = \frac{1}{4} \) each time, you create a predictable sequence. The term 'common' indicates that this difference stays the same no matter which term you move from and to in the sequence.

• In our exercise, every term is formed by adding \( \frac{1}{4} \) to the previous term.
• This ensures the entire sequence can be laid out as: \( \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} \).

Recognizing this pattern simplifies understanding and creating such sequences.

General Formula

The fundamental equation governing arithmetic sequences is the general formula for the n-th term:
\[ a_n = a_1 + (n - 1) \times d, \]where:

• \( a_n \) is any term in the sequence you're interested in
• \( a_1 \) is the first term of the sequence
• \( d \) is the common difference

This formula is a practical tool because it lets you pinpoint any term without having to list every preceding one.
All you need is the first term and the common difference. Then just plug these values into the formula.
When you understand how to use the general formula, you can effortlessly compute any term of the sequence as needed.

N-th Term

The magic of arithmetic sequences is brought to life through the n-th term, which represents any specific location within the sequence.
Using the general formula \( a_n = a_1 + (n - 1) \times d \), we calculate exactly what value will sit at that position.

• You start with the known first term, \( a_1 \).
• Multiply the sequence position minus one, \( (n-1) \), by the common difference, \( d \).
• Add this result to \( a_1 \) to determine \( a_n \).

So, for the sequence given:
If we needed the 5th term, we'd substitute \( n = 5 \) into our formula:
\[ a_5 = \frac{1}{2} + (5 - 1) \times \frac{1}{4} = \frac{1}{2} + 1 = \frac{3}{2} \]
This highlights how the n-th term formula connects any position within the sequence to a precise value.