Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) for each arithmetic sequence. $$S_{28}=2926, a_{28}=199$$

Step-by-Step Solution

Answer

The first term, \(a_1\), is 10.

Step 1: Understand the Formula for the Sum of an Arithmetic Sequence

The sum of the first \(n\) terms of an arithmetic sequence is given by the formula \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( S_n \) is the sum of the first \(n\) terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.

Step 2: Substitute Known Values into the Sum Formula

Substitute \( n = 28 \), \( S_{28} = 2926 \), and \( a_{28} = 199 \) into the formula: \[ 2926 = \frac{28}{2} (a_1 + 199) \].

Step 3: Simplify the Equation

Simplify the equation \( 2926 = 14(a_1 + 199) \) by dividing both sides by 14 to isolate \(a_1 + 199\): \[ 209 = a_1 + 199 \].

Step 4: Solve for \(a_1\)

Subtract 199 from both sides of the equation \( 209 = a_1 + 199 \) to find \(a_1\): \[ 209 - 199 = a_1 \].

Step 5: Final Calculation

Perform the subtraction: \( a_1 = 10 \). Therefore, the first term \(a_1\) is 10.

Key Concepts

Sum of an Arithmetic Sequence

Understanding the sum of an arithmetic sequence is fundamental when working with this particular type of sequence. An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sum of the first \( n \) terms of an arithmetic sequence is calculated using the formula:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

where:

• \( S_n \) represents the sum of the first \( n \) terms.
• \( a_1 \) is the first term.
• \( a_n \) is the nth term.

For the exercise in question, you have parameters: \( n = 28 \), \( S_{28} = 2926 \), and \( a_{28} = 199 \). By substituting these values into the formula, you can solve for the missing term. By understanding how to manipulate and utilize this formula, you can tackle various problems involving arithmetic sequences efficiently.

Finding the First Term

In many arithmetic sequence problems, finding the first term is a key step to understand the sequence as a whole. For instance, you may be given the sum of the sequence and an nth term, such as in the exercise: \[ 2926 = \frac{28}{2} (a_1 + 199) \] Simplifying this is typically the first step. The equation can be reworked by dividing both sides by 14, leading to:\[ 209 = a_1 + 199 \]To isolate \( a_1 \), subtract the given nth term (199) from both sides. This gives:\[ a_1 = 209 - 199 \]Thus, \( a_1 = 10 \). By following these steps methodically, you can determine the first term of the sequence, which is vital for further calculations and understanding the sequence's pattern.

Sequence Formula

To effectively work with any sequence, especially arithmetic sequences, being familiar with the sequence formula is crucial. Arithmetic sequence formulas help identify properties of the sequence and solve for unknowns. The general formula for the \( n \)-th 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.
• \( d \) is the common difference between the terms.

Although in the exercise's solution, the term \( a_{28} \) is already known, understanding the formula is critical to solving for the sequence further should additional information be required or if \( d \) needs to be identified. It allows any sequence to be analyzed or extended as needed dependably.