Problem 40 - Precalculus Mathematics for Calculus

Question

The 20 th term of an arithmetic sequence is \(101,\) and the common difference is \(3 .\) Find a formula for the \(n\) th term.

Step-by-Step Solution

Answer

The formula for the n-th term is \(a_n = 3n + 41\).

Step 1: Understand the Problem

We need to find the formula for the \(n\)th term of an arithmetic sequence. We are given that the 20th term, \(a_{20}\), is 101, and the common difference \(d\) is 3.

Step 2: Write the General Formula for an Arithmetic Sequence

The formula for the \(n\)th term of an arithmetic sequence 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: Utilize Given Information

We know that \(a_{20} = 101\). Plugging into the formula, we have: \[a_{20} = a_1 + (20-1) \cdot 3 = 101.\]

Step 4: Solve for the First Term (a_1)

Simplify the equation from Step 3: \(a_1 + 57 = 101\).Solve for \(a_1\):\[a_1 = 101 - 57 = 44.\]

Step 5: Write the Formula for the n-th Term

Now that we have \(a_1 = 44\) and \(d = 3\), substitute these values into the general formula for an arithmetic sequence:\[a_n = 44 + (n-1) \cdot 3.\]

Step 6: Simplify the Expression

Simplify the equation:\[a_n = 44 + 3n - 3.\]Thus, the simplified formula is:\[a_n = 3n + 41.\]

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial element. It is the constant amount by which each term in the sequence increases or decreases from the previous term.
The common difference is usually represented by the letter \(d\).
Knowing the common difference allows us to predict all terms in the sequence.

• If the common difference is positive, the sequence will steadily increase.
• If it is negative, the sequence will decrease.
• If it is zero, all terms will be the same.

To see it in action, consider your arithmetic sequence where the 20th term is 101, and \(d = 3\).
This "3" tells us that each term is 3 units larger than the previous one.

Nth Term Formula

Knowing the formula for the \(n\)th term allows us to find any term in an arithmetic sequence without listing all numbers down to the one we want.
The formula to identify the \(n\)th term is:\[a_n = a_1 + (n-1) \, \cdot \, d\]Where:

• \(a_n\) is the \(n\)th term.
• \(a_1\) is the first term in the sequence.
• \(d\) is the common difference.

In our case, using the information that the 20th term \(a_{20} = 101\) and \(d = 3\), we first determine \(a_1\), the first term, which was found to be 44.
Substituting these values brings us to the general formula \(a_n = 44 + (n-1) \, \cdot \, 3\).

Sequence Formula

The sequence formula is a simplified expression of the \(n\)th term formula, representing a series in a compact form.
This makes every term in the sequence quickly accessible.
Here's how it works:Starting from the general \(n\)th term formula \(a_n = a_1 + (n-1) \, \cdot \, d\), and given that \(a_1 = 44\) and \(d = 3\), you simplify:\[a_n = 44 + 3n - 3\]Upon further simplification, you get:\[a_n = 3n + 41\]Thus, \(a_n = 3n + 41\) becomes the sequence formula for this particular arithmetic series.
This expression compactly describes every term's position relative to the first term.