Problem 22 - Algebra 2

Question

Find the first five terms of each arithmetic sequence described. \(a_{1}=41, d=5\)

Step-by-Step Solution

Answer

The first five terms are 41, 46, 51, 56, and 61.

Step 1: Understand the Problem

We need to find the first five terms of an arithmetic sequence given the first term \(a_1\) and the common difference \(d\).

Step 2: Use the Formula for an Arithmetic Sequence

The general formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \, d\). We will use this formula to find each of the five terms.

Step 3: Calculate the First Term

The first term is already given as \(a_1 = 41\).

Step 4: Calculate the Second Term

Substitute \(n = 2\) into the formula: \[ a_2 = 41 + (2-1) \times 5 = 41 + 5 = 46 \]

Step 5: Calculate the Third Term

Substitute \(n = 3\) into the formula: \[ a_3 = 41 + (3-1) \times 5 = 41 + 10 = 51 \]

Step 6: Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \[ a_4 = 41 + (4-1) \times 5 = 41 + 15 = 56 \]

Step 7: Calculate the Fifth Term

Substitute \(n = 5\) into the formula: \[ a_5 = 41 + (5-1) \times 5 = 41 + 20 = 61 \]

Step 8: Conclusion: List the First Five Terms

The first five terms of the arithmetic sequence are: 41, 46, 51, 56, and 61.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a key element that sets one term apart from the next. It’s the constant amount that each term increases by to form a sequence. In the problem you’re working on, the common difference is denoted by the letter "d." For our sequence, we have a common difference of \(d = 5\). This means each term in our sequence increases by 5 from the previous one.
Understanding the common difference helps you predict future terms and see how the sequence progresses. It is this consistent increment that makes a sequence "arithmetic."

• Each term = Previous term + Common Difference
• For example, if one term is 41, the next will be 46, because 41 + 5 = 46.

Recognizing the common difference allows us to write and solve sequences efficiently.

n-th Term

The n-th term of an arithmetic sequence refers to any term located at position \(n\) within the sequence. This concept is powerful because it enables you to find any term in the sequence without having to list all the previous terms. The formula used to determine the n-th term is:
\[a_n = a_1 + (n-1) \, d\] This formula requires three components:

• \(a_1\): The first term in the sequence.
• \(n\): The position of the term you are trying to find.
• \(d\): The common difference.

By plugging in these values, you can calculate any term in the sequence quickly. For example, to find the 3rd term when \(a_1 = 41\) and \(d = 5\), substitute \(n = 3\) into the formula, giving you \[a_3 = 41 + (3-1)\times5 = 51.\] This helps you determine terms without laborious calculation.

Sequence Terms

Sequence terms refer to the distinct numbers that make up the arithmetic sequence. Each term is spaced evenly, calculated using both the common difference and the initial term.
In our sequence, the first term \(a_1 = 41\) already sets our starting point. From there, each subsequent term can be calculated by repeatedly adding the common difference. The first five terms in our sequence are:

• 1st term: \(41\)
• 2nd term: \(46\)
• 3rd term: \(51\)
• 4th term: \(56\)
• 5th term: \(61\)

These terms showcase how the arithmetic sequence progresses in a linear and predictable manner, driven by the common difference.

Mathematical Formula

The mathematical formula for an arithmetic sequence is primarily used to find the n-th term, as previously discussed. This formula, \(a_n = a_1 + (n-1) \, d\), is a cornerstone in understanding and working with arithmetic sequences.
The beauty of this formula lies in its simplicity. It allows you to:

• Identify any term if you know the position \(n\).
• See the sequence's growth based on its starting point and its rate (common difference).
• Develop insights into how sequences are structured mathematically.

When applying this formula, ensure you plug in the correct values for \(a_1\), \(n\), and \(d\), to quickly and accurately compute the desired term in the sequence. This formula is essential for navigating and solving problems related to arithmetic sequences.