Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) for each arithmetic sequence. $$S_{3}=75, a_{3}=22$$

Step-by-Step Solution

Answer

\( a_1 = 28 \)

Step 1: Understand the Problem

We are given an arithmetic sequence where the sum of the first three terms is 75, i.e., \( S_3 = 75 \), and the third term of the sequence is 22, i.e., \( a_3 = 22 \). We need to find the first term \( a_1 \) of this sequence.

Step 2: State the Formula for the Sum of an Arithmetic Sequence

The sum of the first \( n \) terms of an arithmetic sequence can be calculated using the formula \( S_n = \frac{n}{2} (a_1 + a_n) \). For \( n = 3 \), this becomes \( S_3 = \frac{3}{2} (a_1 + a_3) \).

Step 3: Substitute Known Values into the Sum Formula

Substitute the known values \( S_3 = 75 \) and \( a_3 = 22 \) into the formula: \[ 75 = \frac{3}{2} (a_1 + 22) \].

Step 4: Solve for \( a_1 \)

To isolate \( a_1 \), first multiply both sides by \( \frac{2}{3} \) to eliminate the fraction: \[ a_1 + 22 = \frac{2}{3} \, \times \, 75 \]. Calculate \( \frac{2}{3} \times 75 = 50 \). Hence, \( a_1 + 22 = 50 \).

Step 5: Find \( a_1 \)

Subtract 22 from both sides to find \( a_1 \): \[ a_1 = 50 - 22 \]. Thus, \( a_1 = 28 \).

Key Concepts

Series Sum Formula

In arithmetic sequences, a series sum formula helps us find the total of the first few terms in the sequence. For an arithmetic sequence, this formula is structured as: \[ S_n = \frac{n}{2} (a_1 + a_n) \] where:

• \( S_n \) is the sum of the first \( n \) terms,
• \( a_1 \) is the first term,
• \( a_n \) is the \( n \)-th term.

This formula simplifies calculating the total of specific terms by utilizing the average of the first and the \( n \)-th term, multiplied by the number of terms divided by two. For example, if you wanted to find the sum of the first three terms \( S_3 \), you would substitute \( n = 3 \) along with the values of \( a_1 \) and \( a_3 \). In this problem, the value \( S_3 = 75 \) was already given, which directly allows us to find \( a_1 \) once \( a_3 \) is known.

First Term of Sequence

Finding the first term \( a_1 \) in an arithmetic sequence is often essential, as it serves as the base from which all other terms build. The formula for sequences means that each term is related, but calculations often begin by establishing \( a_1 \) first. Knowing just the sum of a set of initial terms and a specific term number, we can determine \( a_1 \) efficiently.Here's what you do when you have indirect information like a sum and a later term in the sequence:

• Utilize the series sum formula by inserting known values.
• Rearrange the equation to solve for \( a_1 \) once it is isolated.

In the exercise provided, substituting into the formula gives us: \[ 75 = \frac{3}{2} (a_1 + 22) \] and leads to isolating \( a_1 \) by performing simple arithmetic. This shows that once we've factored out or simplified in between steps, finding the first term becomes a quick task.

Problem-Solving in Mathematics

Problem-solving in mathematics involves breaking complex problems into smaller, manageable parts. Using steps ensures clarity and precision. The arithmetic sequence problem illustrated a clear path from understanding series sums to isolating unknown terms.Here's the detailed breakdown:

• Start by understanding what the problem gives and what it asks. Fully grasp the terms and the relationship between them.
• Use known formulas like the series sum formula, identify the constants provided, and lay out the formula with these known values.
• Break down the equation solving steps to isolate the desired variable, using logical operations like adding, subtracting, multiplying, or dividing.

This structured approach to problem-solving lets students see not only the mathematics but also the logical deductions required for finding a solution, exemplified by finding \( a_1 \). Consistent use of this methodology builds algebraic thinking and boosts confidence in tackling similar problems.