Problem 32 - Precalculus: Mathematics for Calculus

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(7,4,1,-2, \dots\)

Step-by-Step Solution

Answer

The nth term of the sequence is given by \(a_n = 10 - 3n\).

Step 1: Identify the Pattern

Examine the difference between consecutive terms in the sequence: \(7 - 4 = 3\), \(4 - 1 = 3\), \(1 - (-2) = 3\). This consistent difference suggests this is an arithmetic sequence with a common difference \(d\) of \(-3\).

Step 2: Use the Formula for the n-th Term of an Arithmetic Sequence

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

Step 3: Substitute Values into the Formula

Substitute the known values into the formula: \(a_1 = 7\) and \(d = -3\). Thus, the formula becomes: \(a_n = 7 + (n - 1)(-3)\).

Step 4: Simplify the Expression

Simplify the expression: \(a_n = 7 - 3(n - 1)\), which can be expanded to \(a_n = 7 - 3n + 3\).

Step 5: Finalize the Expression

Combine like terms to obtain the final expression: \(a_n = 10 - 3n\). Hence, the explicit formula for the \(n\)th term of the sequence is \(a_n = 10 - 3n\).

Key Concepts

Common Difference

An arithmetic sequence is a set of numbers where the difference between consecutive terms is constant. This difference is called the **common difference** and is often denoted by the letter \(d\).
Understanding this concept is crucial because it helps identify whether a sequence is arithmetic or not.
In the given sequence of numbers, \(7, 4, 1, -2, \ldots\), we observe that:

• The difference between \(7\) and \(4\) is \(-3\)
• The difference between \(4\) and \(1\) is also \(-3\)
• Similarly, the difference between \(1\) and \(-2\) is \(-3\)

So, the common difference \(d\) is \(-3\), indicating that as you progress through the sequence, each term decreases by \(3\). This consistent decrease allows us to confirm the sequence is indeed arithmetic.

nth Term Formula

The **nth term formula** of an arithmetic sequence provides a way to find any term in the sequence without needing to list all previous terms. The formula is:
\[a_n = a_1 + (n - 1) \cdot d\]
Here:

• \(a_n\) represents the nth term of the sequence
• \(a_1\) is the first term, which in our case is \(7\)
• \(n\) indicates the position of the term in the sequence
• \(d\) is the common difference, calculated as \(-3\)

The formula helps to quickly determine the value of any term located at the nth position.
For our sequence, it becomes:
\(a_n = 7 + (n - 1)(-3)\).
Substituting and simplifying yields \(a_n = 10 - 3n\).
So, with this formula, finding terms in the sequence becomes straightforward.

Sequence Patterns

Recognizing **sequence patterns** is key to understanding how sequences are formed and arrayed.
Such recognition enables us to apply the correct mathematical formulas and logic.
In arithmetic sequences, the pattern is established by the common difference and can be expressed linearly.
Each term relates hence systematically to the first term based on that consistent interval.

With the given sequence, pattern recognition involves first identifying the decrease by \(3\).
Then align this with the general arithmetic approach:

• Start with the initial term \(a_1 = 7\)
• Continue by subtracting \(3\) successively (controlled by the common difference)

This recognition enables one to use the sequence's inherent linear pattern to compute any term in it quickly.
Patterns bring predictability, which is the key to solving and understanding sequence problems efficiently.