Problem 38 - Precalculus Mathematics for Calculus

Question

The 12 th term of an arithmetic sequence is \(32,\) and the fifth term is \(18 .\) Find the 20 th term.

Step-by-Step Solution

Answer

The 20th term is 48.

Step 1: Understanding the Problem

We are provided with the 12th and 5th terms of an arithmetic sequence, which are 32 and 18 respectively. We need to find the 20th term of the same sequence. An arithmetic sequence has a common difference, often labeled as \(d\). We also know the general formula for the \(n\)-th term of an arithmetic sequence, \(a_n = a_1 + (n-1)\cdot d\), where \(a_1\) is the first term.

Step 2: Finding the Common Difference

We can use the information given to determine the common difference \(d\). From the equation for the 12th term: \(a_{12} = a_1 + 11d = 32\), and for the 5th term: \(a_5 = a_1 + 4d = 18\). By subtracting the two equations, \((a_1 + 11d) - (a_1 + 4d) = 32 - 18\), we solve for \(d\). This gives us \(7d = 14\), hence \(d = 2\).

Step 3: Finding the First Term

Using the value of \(d = 2\) and the equation for the 5th term \(a_5 = a_1 + 4 \cdot 2 = 18\), we substitute to find \(a_1\). Simplifying gives us \(a_1 + 8 = 18\), so \(a_1 = 10\).

Step 4: Computing the 20th Term

Now that we have both \(a_1 = 10\) and \(d = 2\), we can find the 20th term using the general formula: \(a_{20} = a_1 + 19d = 10 + 19 \cdot 2\). This simplifies to \(a_{20} = 10 + 38 = 48\).

Key Concepts

Common Difference

In an arithmetic sequence, each term is derived by consistently adding a fixed number to the previous term. This fixed number is called the "common difference," represented by \(d\). Calculating the common difference is a vital first step in solving any arithmetic sequence problem.
The formula to find \(d\) involves subtracting the value of an earlier term from a later term and then dividing by the difference in their positions. For instance, with the 12th term \(a_{12} = 32\) and the 5th term \(a_5 = 18\), the common difference is found by:

• Subtracting the two positions: \(12 - 5 = 7\)
• Subtracting their values: \(32 - 18 = 14\)
• Dividing the difference in values by the difference in positions: \(\frac{14}{7} = 2\)

Thus, the common difference for the sequence is \(d = 2\). Understanding and calculating this allows us to find any term in the sequence.

N-th Term Formula

The \(n\)-th term formula for an arithmetic sequence provides a systematic way to find any term within the sequence. The formula is expressed as:
\[ a_n = a_1 + (n-1) \cdot d \]
Where:

• \(a_n\) is the \(n\)-th term we wish to find.
• \(a_1\) is the first term of the sequence.
• \(n\) is the term number.
• \(d\) is the common difference.

This formula organizes all the sequence information neatly, enabling us to find any term once we know the first term and the common difference.
For example, with our values \(a_1 = 10\) and \(d = 2\), we applied it to find the 20th term:
\[ a_{20} = 10 + (20-1) \cdot 2 = 10 + 38 = 48 \]
The \(n\)-th term formula simplifies the process, as we just focus on substituting the given values accordingly.

First Term Determination

In many problems involving arithmetic sequences, knowing the first term \(a_1\) is crucial. It serves as the foundation for applying the \(n\)-th term formula.
When faced with different terms and a common difference, we arrange equations to find \(a_1\). The given problem gives us the 5th term and common difference:
For the 5th term:

• \(a_5 = a_1 + 4 \cdot d = 18\)
• Given that \(d = 2\), we substitute to solve: \(a_1 + 8 = 18\)
• Solving the equation gives: \(a_1 = 10\)

Determining \(a_1\) this way ensures you can incidentally apply the \(n\)-th term formula for any sequence term calculation. It anchors the sequence, confirming how each subsequent term relates back to the starting point.