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

Step-by-Step Solution

Answer

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

Step 1: Understand the formula for an arithmetic sequence

The general formula for an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\), where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. This formula will be used to set up equations for the given terms.

Step 2: Set Up the Equation for the Ninth Term

Using the formula, substitute the known values for the ninth term: \(a_9 = a_1 + (9-1)d = a_1 + 8d = 54\). This gives the first equation: \(a_1 + 8d = 54\).

Step 3: Set Up the Equation for the Seventeenth Term

Similarly, substitute the known values for the seventeenth term: \(a_{17} = a_1 + (17-1)d = a_1 + 16d = 102\). This provides the second equation: \(a_1 + 16d = 102\).

Step 4: Solve the System of Equations

We now have two equations: 1. \(a_1 + 8d = 54\)2. \(a_1 + 16d = 102\)Subtract the first equation from the second: \((a_1 + 16d) - (a_1 + 8d) = 102 - 54\), simplifying gives \(8d = 48\). Solve for \(d\): \(d = \frac{48}{8} = 6\).

Step 5: Substitute Back to Find the First Term

Now that we know \(d = 6\), substitute \(d\) back into the first equation:\(a_1 + 8 \times 6 = 54\), which simplifies to \(a_1 + 48 = 54\). Solving for \(a_1\) gives \(a_1 = 54 - 48 = 6\).

Key Concepts

First Term

The first term of an arithmetic sequence is a crucial element, as it initiates the pattern from which all other terms can be derived. In any arithmetic sequence, the first term, commonly denoted as \(a_1\), sets the stage for the progression by providing a starting reference point.Understanding how to find the first term is key when working with situations where two terms of the sequence are known, and you are tasked with determining the sequence's beginning. To isolate \(a_1\), as outlined in the given problem, you can establish equations using other known terms and their respective positions within the sequence. By substituting these values into the general arithmetic sequence formula, the sequence formula \(a_n = a_1 + (n-1)d\), it's possible to solve for \(a_1\) once the common difference \(d\) is known.In this specific exercise, solving for the first term \(a_1\) involved working through a system of equations based on two given terms, \(a_9=54\) and \(a_{17}=102\). The solution demonstrated how strategic substitution and manipulation of these equations allow you to find that \(a_1 = 6\), which is the critical starting point for this arithmetic sequence.

Common Difference

The common difference in an arithmetic sequence is the consistent amount that each term increases (or decreases) from one term to the next. Denoted by \(d\), it plays a fundamental role in determining the sequence's rate of change.To find the common difference, you can use the equation derived from the known terms. Let's consider the two terms in the sequence from the original problem: the ninth term \(a_9\) and the seventeenth term \(a_{17}\). By setting up equations based on these terms, the difference between the coefficients in these equations reveals \(d\).The step-by-step solution shows how the subtraction of the ninth term equation from the seventeenth term equation simplifies to \(8d = 48\), allowing us to solve for \(d\). Dividing both sides by 8 gives \(d = 6\). This signifies that each term in the sequence is 6 more than the preceding term.This constant difference \(d = 6\) is the backbone of this specific arithmetic sequence, enabling us to project the pattern indefinitely in both directions.

System of Equations

Solving for unknown variables in arithmetic sequences often involves the use of a system of equations. This powerful mathematical tool allows you to find the values of multiple variables by utilizing more than one equation.In the problem provided, two equations were formed from the terms given: \(a_9 = a_1 + 8d = 54\) and \(a_{17} = a_1 + 16d = 102\). These two equations create a system because they have two key unknowns, \(a_1\) and \(d\), with each equation providing different constraints on these variables.The approach involves strategic algebraic manipulation to eliminate one variable, allowing for the isolation and solving of the remaining unknown. In our exercise, subtracting the equation for \(a_9\) from the equation for \(a_{17}\) removed \(a_1\) from the equations, thereby isolating \(d\). Once \(d\) was determined, it was substituted back into one of the equations to solve for the first term \(a_1\).This systematic process shows the efficiency and necessity of systems of equations when analyzing arithmetic sequences, or any situation requiring multiple unknowns to be solved simultaneously.