Problem 3 - Algebra for College Students

Question

Find the general term of the sequence \(-3,-8,-13\), $$ -18, \ldots . $$ $$ -5 n+2 $$

Step-by-Step Solution

Answer

The general term is \(a_n = -5n + 2\).

Step 1: Understand the Problem

We need to determine the general term of an arithmetic sequence given by the first three terms: \(-3, -8, -13\). Each term decreases by a constant amount as we progress.

Step 2: Determine the Common Difference

To find the common difference \(d\) of the sequence, subtract the first term from the second term: \[d = -8 - (-3) = -8 + 3 = -5\]Thus, the common difference \(d = -5\).

Step 3: Use the Formula for the General Term

The nth term \(a_n\) of an arithmetic sequence can be defined using the formula:\[a_n = a_1 + (n-1) imes d\]where \(a_1\) is the first term and \(d\) is the common difference.

Step 4: Substitute the Values into the Formula

Using \(a_1 = -3\) and \(d = -5\), substitute into the formula:\[a_n = -3 + (n-1) imes (-5)\]

Step 5: Simplify the Expression

Expand and simplify the formula:\[a_n = -3 - 5(n - 1)\]\[a_n = -3 - 5n + 5\]\[a_n = -5n + 2\]

Step 6: Verify the General Term

Check the first few terms using \(a_n = -5n + 2\):- For \(n=1\), \(a_1 = -5(1) + 2 = -3\)- For \(n=2\), \(a_2 = -5(2) + 2 = -8\)- For \(n=3\), \(a_3 = -5(3) + 2 = -13\)This matches the given sequence.

Key Concepts

General Term

The general term of an arithmetic sequence is a formula that helps you find any term in the sequence, without having to list out all the previous terms. In this exercise, we're looking at the sequence \(-3, -8, -13, -18, \ldots\). By identifying a pattern or constant difference, we can form a general expression, denoted as \(a_n\), to describe any term in the sequence. This formula is vital as it saves time and effort, especially when dealing with sequences containing many terms or when you need to find terms that are far along in the sequence.​

• The general term acts like a magic formula for a sequence, letting you jump directly to your desired term.
• It's particularly helpful when sequences have a predictable pattern, like arithmetic sequences which have a constant difference.

To find the general term of the sequence, we need just two pieces of information: the first term and the common difference, which will be explained next.

Common Difference

A common difference in an arithmetic sequence is a fixed value that separates consecutive terms. In simpler terms, it's the amount you either add or subtract to move from one term to the next. For our sequence \(-3, -8, -13, \ldots \), the common difference \(d\) is calculated by finding the difference between any two successive terms. For example, \(d = -8 - (-3) = -5\). This tells us that each term is 5 less than the term before it.

• Understanding this difference helps not only in identifying the nature of the sequence but also in forming the general term.
• Knowing this difference is key for predictions about any term in the sequence.

Whenever sequences are involved, always look for a consistent difference or pattern to understand the sequence better.

Nth Term Formula

The nth term formula for arithmetic sequences can be represented as:\[a_n = a_1 + (n-1) \times d\]where:

• \(a_n\) represents the nth term you want to find.
• \(a_1\) is the first term of the sequence, which is \(-3\) in this case.
• \(d\) is the common difference, calculated as \(-5\).

This formula provides a direct path to calculating any term in the sequence without iterative counting. By substituting, we derive:\[a_n = -3 + (n-1) \times (-5)\]Solving, we simplify to:\[a_n = -5n + 2\].This gives the compact expression to find any term of the sequence.

Sequence Verification

Verification in sequences is the process of checking if the derived general term formula correctly generates the sequence's specific terms. Essentially, it's about double-checking the math to ensure the formula accurately represents the sequence's pattern.For the given general term \(a_n = -5n + 2\), let's verify:

• For \(n=1\), \(a_1 = -5(1) + 2 = -3\)
• For \(n=2\), \(a_2 = -5(2) + 2 = -8\)
• For \(n=3\), \(a_3 = -5(3) + 2 = -13\)

Each value matches the terms of the original sequence, confirming our formula works. Verification acts as a quality check, ensuring your calculations accurately mirror the sequence’s structure. Always make a habit of verifying your results in math to minimize errors and build confidence in your solutions.