Problem 25 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) for each arithmetic sequence. $$a_{5}=27, a_{15}=87$$

Step-by-Step Solution

Answer

The first term, \(a_1\), is 3.

Step 1: Understand the Arithmetic Sequence Formula

An arithmetic sequence is defined by the formula for the nth term: \(a_n = a_1 + (n-1)d\). Here, \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference between consecutive terms.

Step 2: Set Up Equations for Known Terms

Use the given terms to set up equations. For \(a_5\), we have the equation \(a_1 + 4d = 27\). For \(a_{15}\), we have \(a_1 + 14d = 87\). These will help us find \(a_1\) and \(d\).

Step 3: Solve the System of Equations

To find \(d\), subtract the first equation from the second: \((a_1 + 14d) - (a_1 + 4d) = 87 - 27\). Simplifying gives \(10d = 60\), which means \(d = 6\).

Step 4: Substitute to Find \(a_1\)

Substitute \(d = 6\) back into one of the equations to solve for \(a_1\). Using \(a_1 + 4d = 27\), substitute for \(d\): \(a_1 + 4(6) = 27\). This simplifies to \(a_1 + 24 = 27\), so \(a_1 = 3\).

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a fundamental element. It represents the constant amount that each term increases or decreases from the previous term. Understanding this helps in identifying the pattern within the sequence.

For an arithmetic sequence defined as \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference, you can use the common difference to predict future terms once you know any specific term. For example, in the given problem with terms \(a_5 = 27\) and \(a_{15} = 87\), the common difference \(d\) was derived from the subtraction and solving of the system of equations to be \(d = 6\). This means each term is 6 units larger than the term before it.

When trying to find patterns or determine any particular term's value, identifying the common difference is a crucial first step.

System of Equations

Solving a system of equations is a powerful method used in various mathematical situations, including finding unknowns in arithmetic sequences. By using known terms and their corresponding equations, we can derive values for both the common difference and the first term of a sequence.

Let's break it down for our example with \(a_5 = 27\) and \(a_{15} = 87\):

• We set up the first equation as \(a_1 + 4d = 27\) for \(a_5\).
• The second equation is \(a_1 + 14d = 87\) for \(a_{15}\).

By subtracting the first equation from the second, we eliminate \(a_1\). This results in the equation \(10d = 60\), which simplifies to \(d = 6\).

With the common difference known, it becomes straightforward to substitute back into any of the original equations to find the first term.

First Term of a Sequence

The first term of an arithmetic sequence is often denoted as \(a_1\). It is the starting point of the sequence from which all other terms are derived. Knowing \(a_1\) along with the common difference \(d\), allows you to construct the entire sequence.

In this particular example, after identifying the common difference was \(d = 6\), we return to the equation \(a_1 + 4(6) = 27\). By solving for \(a_1\), you subtract 24 from both sides, resulting in \(a_1 = 3\).

This first term, \(a_1 = 3\), sets the foundation for our sequence, confirming the formula \(a_n = 3 + (n-1) \, 6\) will accurately compute terms of this sequence. It's the key to understanding the initial value of the sequence's progression.