Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) for each arithmetic sequence. $$S_{20}=-1300, a_{20}=-122$$

Step-by-Step Solution

Answer

The first term of the sequence is \( a_1 = -8 \).

Step 1: Understand the problem

We are given an arithmetic sequence with the sum of the first 20 terms ( S_{20} = -1300 ) and the 20th term ( a_{20} = -122 ). We need to find the first term ( a_1 ) of the sequence.

Step 2: Use 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: \[ S_n = \frac{n}{2} (a_1 + a_n) \] Given that S_{20} = -1300 and a_{20} = -122 , we will plug these values into the formula to solve for a_1 .

Step 3: Substitute known values into the sum formula

Plug the known values into the sum formula: \[ -1300 = \frac{20}{2} (a_1 + (-122)) \] Simplify: \[ -1300 = 10(a_1 - 122) \]

Step 4: Solve for \( a_1 \)

First, divide both sides of the equation by 10: \[ -130 = a_1 - 122 \] Next, add 122 to both sides to isolate a_1 : \[ a_1 = -130 + 122 \] Simplify the result: \[ a_1 = -8 \]

Key Concepts

Sum of Arithmetic Sequence

The arithmetic sequence is a series of numbers where each term increases by a constant difference from the previous term. **Finding the sum of an arithmetic sequence** involves adding all terms up to a certain point. This sum can be effectively calculated using the formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \] where:

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

This formula simplifies the process of summing multiple terms by multiplying the average of the first and nth term by the number of terms. It is especially helpful for large sequences where calculating each term individually would be time-consuming. In our example, given \( S_{20} = -1300 \) and \( a_{20} = -122 \), this formula allows us to solve for unknown terms efficiently.

First Term of Arithmetic Sequence

To find the first term, or \( a_1 \), of an arithmetic sequence, you can rearrange the sum formula to isolate \( a_1 \). Using the known values in the sequence is pivotal to solving the equation. Referring to our example, we know \( S_{20} = -1300 \) and \( a_{20} = -122 \). By substituting these into the formula:\[ -1300 = \frac{20}{2} (a_1 - 122) \] we simplify: \[ -1300 = 10(a_1 - 122) \] This step is important because it reduces the equation making it easier to solve for \( a_1 \). Dividing both sides by 10, we isolate \( a_1 \):\[ -130 = a_1 - 122 \] Finally, add 122 to both sides to find \( a_1 \): \[ a_1 = -8 \] This shows the importance of understanding the role of each term within the formula and how to manipulate the equation to find the unknown terms.

Arithmetic Sequence Formula

The arithmetic sequence formula is fundamental in understanding how sequences are structured. It provides a way to find any term in the sequence if the first term and the common difference are known. Every arithmetic sequence can be expressed using the formula:\[ a_n = a_1 + (n-1)d \] where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term,
• \( n \) is the term number,
• and \( d \) is the common difference.

This formula captures the essence of how each successive term is generated by adding the consistent difference \( d \) to the preceding term starting from \( a_1 \). By using this insight along with the information given in the problem—such as the value of the nth term—it's possible to deduce additional terms or verify the sequence's structure. In the context of our problem, knowing \( a_{20} = -122 \) facilitated our extraction of the first term through reverse calculation.