Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$3,-2,-7,-12, \dots$$

Step-by-Step Solution

Answer

The common difference is \(-5\).

Step 1: Identify the Sequence

The arithmetic sequence given is: \(3, -2, -7, -12, \dots\). This sequence is defined by a starting number and a common difference that is added to each term to get the next term.

Step 2: Determine the Formula of an Arithmetic Sequence

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

Step 3: Find the Common Difference

The common difference \(d\) can be found by subtracting any term from the previous term. For example, \(-2 - 3 = -5\). Similarly, check the next pair: \(-7 - (-2) = -5\). Thus, the common difference is consistent and is \(d = -5\).

Key Concepts

Common Difference

An arithmetic sequence follows a specific pattern where the difference between any two consecutive terms remains constant. This consistent interval is known as the 'common difference', denoted by the letter \(d\). In the sequence provided, which is \(3, -2, -7, -12, \dots\), the common difference \(d\) determines how much each term decreases or increases from the previous one.
To find \(d\), you simply subtract the first term from the second term, the second from the third, and so on. For the sequence provided:

• \(-2 - 3 = -5\)
• \(-7 - (-2) = -5\)
• \(-12 - (-7) = -5\)

Since each result is \(-5\), the common difference is consistent throughout the sequence, confirming that \(d = -5\). This number tells us that each term is 5 less than the preceding term.

Sequence Formula

The structure of an arithmetic sequence can be represented using a sequence formula, which is handy for identifying any term in the sequence without listing all preceding terms. The sequence formula is given by:
\[a_n = a_1 + (n-1) \cdot d\] In this equation:

• \(a_n\) represents the \(n\)-th term of the sequence.
• \(a_1\) is the first term in the sequence.
• \(d\) is the common difference between terms.
• \(n\) stands for the term number.

For instance, if you start with \(a_1 = 3\) and know \(d = -5\), you can find any term in the sequence by plugging these values into the formula. This formula is quite powerful because it allows direct navigation to any term, bypassing potential computation errors.

n-th Term

The concept of the \(n\)-th term is essential for arithmetic sequences. It allows us to express any term in the sequence directly based on its position, \(n\). Instead of manually subtracting repeatedly, you can use the sequence formula:
\[a_n = a_1 + (n-1) \cdot d\]Consider a sequence that begins with \(3\) and has a common difference \(-5\). Suppose you want the 5th term:

• Identify \(a_1 = 3\) and \(d = -5\).
• Plug into the formula: \[a_5 = 3 + (5-1) \cdot (-5)\].
• Calculate: \[a_5 = 3 - 20 = -17\].

Therefore, the 5th term \(a_5\) is \(-17\). This process shows how to calculate any term number quickly and systematically without needing to extend the sequence step-by-step.