Problem 23 - Algebra 2

Question

Find the first five terms of each arithmetic sequence described. \(a_{1}=6, d=-4\)

Step-by-Step Solution

Answer

The first five terms are: 6, 2, -2, -6, -10.

Step 1: Understanding the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This means that to find the next term in the sequence, we add the common difference to the last term we have.

Step 2: Formula for the nth Term

The general formula for finding the nth 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 3: Calculate the First Term

Given \( a_1 = 6 \), the first term of our sequence is 6. Thus, \( a_1 = 6 \).

Step 4: Calculate the Second Term

To find the second term \( a_2 \), use the formula:\( a_2 = a_1 + (2-1) imes d \) \( a_2 = 6 + 1 imes (-4) = 6 - 4 = 2 \)

Step 5: Calculate the Third Term

To find the third term \( a_3 \), use the formula:\( a_3 = a_1 + (3-1) imes d \)\( a_3 = 6 + 2 imes (-4) = 6 - 8 = -2 \)

Step 6: Calculate the Fourth Term

To find the fourth term \( a_4 \), use the formula:\( a_4 = a_1 + (4-1) imes d \) \( a_4 = 6 + 3 imes (-4) = 6 - 12 = -6 \)

Step 7: Calculate the Fifth Term

To find the fifth term \( a_5 \), use the formula:\( a_5 = a_1 + (5-1) imes d \) \( a_5 = 6 + 4 imes (-4) = 6 - 16 = -10 \)

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the common difference, usually denoted as \(d\), plays a crucial role. This difference is the fixed amount added (or subtracted) to one term to get the next term.
When talking about sequences, it’s all about regularity and repetition. For instance, in the sequence you are looking at, the first term is 6, and the common difference is \(-4\). To find the next term, you simply add this common difference to the current term.
Here’s a quick step:

• Start with 6 (the first term).
• Add \(-4\) to 6 to get the second term, which is 2.
• Keep adding \(-4\) to each new term to find the subsequent terms.

As a sequence progresses, this common difference remains constant, driving the uniform spread of terms throughout the sequence.

Exploring the nth Term Formula

The nth term formula is like a key that unlocks the terms of an arithmetic sequence. It helps you find any term without having to list all preceding terms. The formula is expressed as:
\[a_n = a_1 + (n-1) \times d\]
Where:

• \(a_n\) is the term you want to find.
• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference.

For example, if you want to find the 3rd term of the sequence, you plug in the values like this:
\[a_3 = 6 + (3-1) \times (-4)\]
This simplifies to \(6 - 8 = -2\). Use this formula for any position \(n\) you desire.

Grasping Sequence Terms

Sequence terms are the individual numbers that make up the sequence. In an arithmetic sequence, these terms are aligned neatly and follow a predictable pattern due to their common difference.
The first five terms of the sequence given are 6, 2, -2, -6, and -10.
Why are these specific numbers important? Each term is a step along the path determined by the starting point (6) and how far you jump each time (-4, the common difference).
Think of it as walking down a path: starting at point 6, each step back measures \(-4\), leading you from one term to the next. Knowing the sequence terms helps paint a full picture of the sequence's progression.