Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, find the first term given two terms from an arithmetic sequence. Find the first term or \(a_{1}\) of an arithmetic sequence if \(a_{11}=11\) and \(a_{21}=16\).

Step-by-Step Solution

Answer

The first term is 6.

Step 1: Understand the Problem

We need to find the first term of an arithmetic sequence given two terms: the 11th term, which is 11, and the 21st term, which is 16.

Step 2: Identify the Formula for the n-th Term of 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: Create Equations Based on Given Terms

Using the formula, create equations based on the information given: For the 11th term: \[ a_{11} = a_1 + 10d = 11 \] For the 21st term: \[ a_{21} = a_1 + 20d = 16 \]

Step 4: Solve the System of Equations for Common Difference \(d\)

Subtract the equation for \(a_{11}\) from the equation for \(a_{21}\): \[ (a_1 + 20d) - (a_1 + 10d) = 16 - 11 \] \[ 10d = 5 \] Solving for \(d\), we get: \[ d = \frac{5}{10} = \frac{1}{2} \]

Step 5: Solve for the First Term \(a_1\)

Substitute \(d = \frac{1}{2}\) back into the equation for \(a_{11}\): \[ a_1 + 10 \cdot \frac{1}{2} = 11 \] \[ a_1 + 5 = 11 \] Solving for \(a_1\), we get: \[ a_1 = 11 - 5 = 6 \]

Step 6: Conclusion: State the First Term

The first term \(a_1\) of the arithmetic sequence is 6.

Key Concepts

First Term

In any arithmetic sequence, the first term is crucial as it sets the starting point for the rest of the sequence. It's commonly denoted by \(a_1\).

Think of \(a_1\) as the base from which all subsequent terms are built. By knowing \(a_1\), we can easily predict or calculate any other term within this sequence using the formula for the n-th term.

For example, in many problems, you are given other terms from the sequence, like the 11th or 21st terms, and need to work backward to find \(a_1\). Doing this requires understanding the sequence's structure and using given information effectively.

Common Difference

The common difference, denoted by \(d\), is a consistent interval between any two consecutive terms in an arithmetic sequence. You can think of it as the 'step' you need to reach the next term.

To find the common difference, you solve equations based on known terms of the sequence. For instance, using given information like \(a_{11} = 11\) and \(a_{21} = 16\), you can form the equations:

• \(a_{11} = a_1 + 10d = 11\)
• \(a_{21} = a_1 + 20d = 16\)

Subtract these equations to isolate \(d\), resulting in 10 terms of \(d\) equaling 5. When solved, \(d = \frac{1}{2}\), meaning each step from one term to the next is by half.

n-th Term Formula

The n-th term formula is a powerful tool in assessing an arithmetic sequence. The formula is expressed as \(a_n = a_1 + (n-1) \cdot d\). This tells us the value of any term in the sequence based on its position \(n\), the first term \(a_1\), and the common difference \(d\).

This formula is important because it enables predictions and calculations of unknown terms without listing all terms of the sequence. For example, if you wanted the 11th term, you'd plug the numbers into the formula to find it effortlessly.

Such formulas help in quickly resolving sequence-related questions, making \(a_n = a_1 + (n-1) \cdot d\) an indispensable part of arithmetic computations.