Problem 22 - 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 \(a_{9}=54\) and \(a_{17}=102 .\)

Step-by-Step Solution

Answer

The first term \(a_1\) is 6.

Step 1: Understanding the Problem

We are given two terms of an arithmetic sequence: the ninth term \(a_9 = 54\) and the seventeenth term \(a_{17} = 102\). We are asked to find the first term \(a_1\). In an arithmetic sequence, the difference between consecutive terms is constant and is denoted as \(d\). The \(n\)-th term of an arithmetic sequence can be expressed as \(a_n = a_1 + (n-1)d\).

Step 2: Express the Given Terms Using the Formula

Using the formula for the \(n\)-th term, express the given terms:\(a_9 = a_1 + 8d = 54\)\(a_{17} = a_1 + 16d = 102\).

Step 3: Set Up a System of Equations

We have two equations based on the given terms:\(1) \; a_1 + 8d = 54\)\(2) \; a_1 + 16d = 102\).

Step 4: Solve for \(d\)

Subtract equation (1) from equation (2) to eliminate \(a_1\) and solve for \(d\):\((a_1 + 16d) - (a_1 + 8d) = 102 - 54\)\(16d - 8d = 48\)\(8d = 48\)\(d = 6\).

Step 5: Solve for \(a_1\)

Use the value of \(d = 6\) in one of the equations to find \(a_1\). Substituting \(d = 6\) into equation (1):\(a_1 + 8(6) = 54\)\(a_1 + 48 = 54\)\(a_1 = 54 - 48\)\(a_1 = 6\).

Step 6: Conclusion

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

Key Concepts

First Term

In an arithmetic sequence, the first term, often denoted as \(a_1\), is the starting point from which the sequence begins. It's fundamental in determining how the sequence unfolds. Think of the first term as the anchor that dictates where your sequence starts. For example, in our exercise, we're tasked with finding this first term given two other terms in the sequence.The importance of the first term cannot be overstated: without it, you can't use the sequence formula effectively. The first term helps set the initial conditions for the entire sequence. When you know \(a_1\), calculations for other terms in the sequence become straightforward.

Common Difference

In an arithmetic sequence, the "common difference," commonly represented as \(d\), is what differentiates an arithmetic sequence from other types of sequences. It's a constant value that you add to each term to get to the next one. Thus, every term in the sequence can be derived by adding this common difference to the previous term.

• This consistent addition is what makes the sequence "arithmetic."
• If \(d\) is positive, the sequence increases.
• If \(d\) is negative, the sequence decreases.

In the exercise, we calculate \(d\) using two given terms by setting up equations and subtracting them to find \(d = 6\). Using this approach allows us to move step by step in understanding how the sequence grows or shrinks consistently.

System of Equations

A system of equations is a set of equations with the same variables. In the context of arithmetic sequences, you can use a system of equations to solve for unknowns like the first term or the common difference. For instance, when you are given two terms of an arithmetic sequence, you can set up two equations to find \(a_1\) and \(d\).When tackling these problems:

• Always begin by applying the sequence formula \(a_n = a_1 + (n-1)d\) to the provided terms.
• Equate two expressions involving the same unknown to form equations.
• Subtract one equation from another to eliminate one variable.

This method simplifies the problem, leading you to uncover the values of the first term and the common difference easily.

Sequence Formula

The arithmetic sequence formula \(a_n = a_1 + (n-1)d\) is the backbone of solving any arithmetic sequence problem. This formula helps us find any term in the sequence once we know the first term and the common difference.

• \(a_n\) denotes the nth term you want to find.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the specific position of the term in the sequence.

In our exercise, the formula enables us to express the known terms \(a_9 = 54\) and \(a_{17} = 102\) as equations in terms of \(a_1\) and \(d\). This process is what sets the foundation for identifying the unknowns in the given problem. By mastering this formula, you open the door to successfully navigating through any arithmetic sequence challenge.