Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) for each arithmetic sequence. $$S_{16}=-160, a_{16}=-25$$

Step-by-Step Solution

Answer

The first term \(a_1\) is 5.

Step 1: Understand the Given Information

We are given an arithmetic sequence where the 16th term \(a_{16} = -25\) and the sum of the first 16 terms \(S_{16} = -160\). We need to find the first term \(a_1\).

Step 2: Recall the Arithmetic Sequence Formula

The formula for the nth term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \, d\), where \(d\) is the common difference. The sum of the first n terms is given by \(S_n = \frac{n}{2} \times (a_1 + a_n)\).

Step 3: Substitute into Sum Formula

Substitute the given values into the sum formula:\[ -160 = \frac{16}{2} \times (a_1 + (-25)) \]Simplify the calculation:\[ -160 = 8 \, (a_1 - 25) \]

Step 4: Solve for \(a_1\)

First, divide both sides by 8 to isolate \(a_1 - 25\):\[ -20 = a_1 - 25 \]Add 25 to both sides to find \(a_1\):\[ a_1 = -20 + 25 \]\[ a_1 = 5 \]

Step 5: Verify the Solution

To verify, substitute \(a_1 = 5\) back into the sum formula and check if it satisfies the conditions set by \(S_{16}\) and \(a_{16}\):Calculate \(d\) using \(a_{16} = -25\), then check calculations to confirm \(S_{16} = -160\) is consistent.

Key Concepts

Sum of Sequences

Calculating the sum of an arithmetic sequence is straightforward once you understand the formula. To find the sum of the first n terms in an arithmetic sequence, we use the formula:

• \( S_n = \frac{n}{2} \times (a_1 + a_n) \)

Here, \(S_n\) is the sum of the first n terms, \(a_1\) is the first term, and \(a_n\) is the nth term. This formula helps gather all terms' values together to provide a total. It's efficient because it calculates the sum using just the first and last term, avoiding the need to add each term individually.
In this specific problem, we used this formula to check that with the first term \(a_1\) being 5, we got the correct sum of -160 for the first 16 terms.

Nth Term Formula

Understanding the nth term formula is crucial in solving problems relating to arithmetic sequences. The formula is:

• \( a_n = a_1 + (n-1) \cdot d \)

where \(a_n\) represents the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference between terms. This formula allows you to find any term in the sequence provided you know the first term and the common difference.
In the exercise, we are given \(a_{16} = -25\). We used this formula to back-calculate \(d\) after finding \(a_1\). It’s a handy method to tackle sequence problems whenever you need to identify a specific term.

Common Difference

The common difference is one of the key properties of arithmetic sequences. It represents the amount by which each term increases or decreases from the previous term. This value is constant across the sequence.

• To calculate the common difference \(d\), you can rearrange the nth term formula to \( d = \frac{a_n - a_1}{n-1} \).

Determining this difference is pivotal because it allows you to fully describe the sequence and predict subsequent terms. In the original problem, although the common difference wasn’t explicitly mentioned, understanding that finding it helps ensure the accuracy of the sequence was vital. With \(a_{16} = -25\) and \(a_1 = 5\), by calculating \(d\), we verify that the entire sequence formulation is correct.

Sequence Verification

Once you have proposed an answer, it’s always smart to verify the sequence. Verification not only checks your arithmetic but also ensures that all conditions set by a problem are met. For sequence problems:

• First, apply the nth term formula to ensure that the calculated terms comply with the given information.
• Second, re-evaluate the sum using the calculated values to confirm they produce the stipulated result.

In our problem, with \(a_1 = 5\), we verified that imposing this value into both the nth term and sum formulas consistently met the conditions \(a_{16} = -25\) and \(S_{16} = -160\). By revisiting calculations and confirming all given data aligns, sequence verification imparts a double-check of accuracy and insight into the sequence's structural integrity.