Problem 40 - Contemporary Precalculus

Question

Use the given information about the arithmetic sequence with common difference d to find a and a formula for \(a_{n}\). $$a_{5}=-3, a_{9}=-18$$

Step-by-Step Solution

Answer

Answer: The first term of the arithmetic sequence is 12. The general formula for the nth term is \(a_n = 12 + (n - 1)(-\frac{15}{4})\).

Step 1: Find the Common Difference (d)

Since we know the values of \(a_{5}\) and \(a_{9}\), we can find the common difference as follows: $$a_9 - a_5 = (a_5 + 4d) - a_5$$ $$-18 - (-3) = 4d$$ $$-15 = 4d$$ $$d = -\frac{15}{4}$$

Step 2: Determine the First Term (a)

Now that we have the common difference d, we can use the arithmetic sequence formula (\(a_n = a + (n - 1)d\)) to find the first term 'a' by replacing \(a_5\) by a value and n by 5: $$a_5 = a + (5 - 1) \times d$$ $$-3 = a + 4(-\frac{15}{4})$$ $$-3 = a - 15$$ $$a = -3 + 15$$ $$a = 12$$

Step 3: Write the General Formula for \(a_{n}\)

Now, as we have the first term a and the common difference d, we can write the general formula for \(a_{n}\) using arithmetic sequence formula: $$a_n = a + (n - 1)d$$ $$a_n = 12 + (n - 1)(-\frac{15}{4})$$ Therefore, the first term of the arithmetic sequence is 12, and the general formula for the nth term is: $$a_n = 12 + (n - 1)(-\frac{15}{4})$$

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial concept. It is a fixed amount added to each term of the sequence to get the next term. This difference remains constant across the sequence. We designate this difference with the symbol \(d\).

• In our example, we found that the sequence values at certain positions, such as \(a_5 = -3\) and \(a_9 = -18\), allowed us to calculate the common difference.
• The formula to find \(d\) is typically \(a_{n+k} - a_n = kd\), where \(k\) is the difference in position numbers of two known terms.
• For this arithmetic sequence, we solved \(-18 - (-3) = 4d\), leading us to find that \(d = -\frac{15}{4}\).

Understanding the common difference helps us unravel the entire sequence, identifying the pattern and making further predictions about other terms.

General Formula

The general formula of an arithmetic sequence is a powerful tool. It helps us find any term within the sequence without knowing all the preceding terms. The formula is generally expressed as: \[a_n = a + (n - 1)d\]Where:

• \(a_n\) is the nth term we are calculating.
• \(a\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) represents the position of the term in the sequence.

For the given sequence, once we knew \(a = 12\) and \(d = -\frac{15}{4}\), we could write the formula:\[a_n = 12 + (n - 1)(-\frac{15}{4})\]This formula allows calculation of any term directly, like \(a_1\), \(a_2\), etc. Just replace \(n\) with the desired position number.

First Term

The first term of an arithmetic sequence, often denoted as \(a\), serves as the initial point of the sequence. It is the term from which the sequence begins and to which the common difference is added successively.

• For our case, we used a known term and the common difference to find the first term. By substituting in the sequence formula, \(a_n = a + (n - 1)d\), and using \(a_5 = -3\), we were able to solve for \(a\).
• The calculation was: \(-3 = a + 4(-\frac{15}{4})\), leading to \(a = 12\).
• Once we find \(a\), it provides a firm base to generate the complete sequence by applying the common difference.

The first term is foundational, setting the starting point for all subsequent computations in the sequence.