Problem 23 - College Algebra

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 $$ a_{11}=11 \text { and } a_{21}=16 $$

Step-by-Step Solution

Answer

The first term \( a_1 \) is 6.

Step 1: Understand the arithmetic sequence formula

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

Step 2: Set up the equations

We know the 11th term \( a_{11} = 11 \) and the 21st term \( a_{21} = 16 \). Using the formula, we set up the following equations:\[ a_{11} = a_1 + 10d = 11 \]\[ a_{21} = a_1 + 20d = 16 \]

Step 3: Subtract equations to find the common difference

Subtract the first equation from the second to eliminate \( a_1 \) and solve for \( d \):\[ (a_1 + 20d) - (a_1 + 10d) = 16 - 11 \]\[ 10d = 5 \]\[ d = \frac{5}{10} = 0.5 \]

Step 4: Solve for the first term \(a_1\)

Substitute \( d = 0.5 \) back into either equation. Using \( a_{11} = a_1 + 10d = 11 \):\[ a_1 + 10(0.5) = 11 \]\[ a_1 + 5 = 11 \]\[ a_1 = 11 - 5 \]\[ a_1 = 6 \]

Step 5: Verify the solution

Verify the solution by substituting \( a_1 = 6 \) and \( d = 0.5 \) into both original equations to check if they hold true. For \( a_{11} = 11 \):\[ a_{11} = 6 + 10(0.5) = 11 \]. For \( a_{21} = 16 \):\[ a_{21} = 6 + 20(0.5) = 16 \]. Both are correct.

Key Concepts

Common Difference

The common difference is a crucial element of an arithmetic sequence, defining the constant interval between consecutive terms. In arithmetic sequences, each term increases or decreases by this same amount, making it a central aspect of their structure. To find the common difference (\(d\)), you subtract any two successive terms. For instance, with the two equations given in the problem: \(a_{11} = a_1 + 10d = 11\) and \(a_{21} = a_1 + 20d = 16\), subtract the first from the second:

• \((a_1 + 20d) - (a_1 + 10d) = 16 - 11\)
• Resulting in \(10d = 5\)

From here, solving gives \(d = 0.5\). This commonality of difference ensures a predictable and consistent pattern across the sequence.

Nth Term Formula

In arithmetic sequences, the way to identify or calculate any term's position is through the nth term formula, articulated as \(a_n = a_1 + (n - 1)d\). This formula hinges on knowing at least three variables to find the fourth.
With the provided values like in our exercise, the same formula transforms respective term positions \(a_{11}\) and \(a_{21}\) into equations that, once manipulated, reveal unknowns such as the common difference or even the first term.
Ultimately, it's a pivotal tool for sequence challenges—ensuring that, once the common difference \(d\) and first term \(a_1\) are found, predicting other terms becomes straightforward.

First Term Calculation

The first term, denoted as \(a_1\), sets the starting point for an arithmetic sequence. Determining it requires manipulation of known sequence terms and the common difference.
Once you've calculated the common difference, substitute it back into any known term equation to solve for \(a_1\).

• Thus, using \(a_{11} = a_1 + 10d = 11\) and substituting \(d = 0.5\):
• \(a_1 + 10(0.5) = 11\)
• Simplifies to \(a_1 + 5 = 11\)
• Finally, \(a_1 = 11 - 5\), thus \(a_1 = 6\)

Thus, calculating the first term from sequence data fosters the foundation for decoding the entirety of an arithmetic sequence.

Sequence Equations

Sequence equations in arithmetic sequences often include multiple terms represented with the nth term formula to gain insights into overall patterns.
In practical terms, each equation represents a snapshot of the sequence at a certain position, and collectively, they help solve for unknown variables like the common difference or the initial term.

• For example, from the problem, deriving two equations \(a_{11} = a_1 + 10d = 11\) and \(a_{21} = a_1 + 20d = 16\), helps illustrate the arithmetic sequence's structure.
• By manipulating or subtracting these equations, it's possible to isolate and solve for key elements of the sequence, reflecting on how arithmetic sequences obey predictable laws.

Thus, sequence equations are indispensable tools for fully understanding and deciphering arithmetic progressions.