Problem 5 - Essential Calculus Early Transcendentals

Question

\(5-8=\) Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. $$\left\\{-3,2,-\frac{4}{3}, \frac{8}{9},-\frac{16}{27}, \ldots\right\\}$$

Step-by-Step Solution

Answer

\(a_{n} = -3\left(-\frac{2}{3}\right)^{n-1}\)

Step 1: Observe the Pattern

Look at the sequence: \(-3, 2, -\frac{4}{3}, \frac{8}{9}, -\frac{16}{27}, \ldots\). Notice that the signs alternate and the absolute values suggest a geometric progression.

Step 2: Identify the Initial Term

The first term of the sequence is \(-3\), which can be rewritten as \(-\frac{3}{1}\).

Step 3: Examine Common Ratio

Calculate the ratio between consecutive terms: \(\frac{2}{-3}, \frac{-\frac{4}{3}}{2}, \ldots\). Simplified, each ratio is \(-\frac{2}{3}\). Thus, the common ratio is \(-\frac{2}{3}\).

Step 4: Construct the General Term Formula

The general term for a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the initial term and \(r\) is the ratio. Using \(a_1 = -3\) and \(r = -\frac{2}{3}\), the formula becomes \(a_n = -3\left(-\frac{2}{3}\right)^{n-1}\).

Step 5: Verify the Formula

Check the formula against initial terms: For \(n=1\), \(a_1 = -3\); for \(n=2\), \(a_2 = 2\); and so forth. They match the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key concept. It is the factor by which we multiply each term to get the next term in the sequence. Identifying this ratio is crucial as it allows us to describe the entire sequence using just one expression. In our example sequence \(-3, 2, -\frac{4}{3}, \frac{8}{9}, -\frac{16}{27}, \ldots\), the common ratio can be found by dividing any term by its previous term. For instance:

• \(\frac{2}{-3} = -\frac{2}{3}\)
• \(\frac{-\frac{4}{3}}{2} = -\frac{2}{3}\)

Every ratio results in \(-\frac{2}{3}\), showing the pattern in which each term is consistently multiplied by \(-\frac{2}{3}\) to get the next term. This consistent ratio helps us determine the progression of terms.

Alternating Signs

A noticeable feature in the sequence is the alternating pattern of signs. This means that the sequence switches from negative to positive, and vice versa, from one term to the next. Alternating signs occur due to multiplying by a negative common ratio. Since our common ratio is \(-\frac{2}{3}\), on multiplying each term by this negative, the sign changes. Such a pattern makes the sequence more complex and adds a rich layer of understanding:

• When multiplied by \(-\frac{2}{3}\), a negative term becomes positive.
• When multiplied by \(-\frac{2}{3}\), a positive term becomes negative.

In essence, the negative common ratio is responsible for flipping the sign of each successive term in the sequence.

General Term Formula

The general term formula in a geometric sequence gives us a way to find any term in the sequence without listing all previous ones. It is expressed as:\[a_n = a_1 \cdot r^{n-1}\]where:

• \(a_n\) is the general term.
• \(a_1\) is the first term in the sequence.
• \(r\) is the common ratio.
• \(n\) is the position of the term.

In our case, for the sequence:\[-3, 2, -\frac{4}{3}, \frac{8}{9}, -\frac{16}{27}, \ldots\]we identified:

• \(a_1 = -3\)
• \(r = -\frac{2}{3}\)

Applying these values, our general term formula becomes:\[a_n = -3\left(-\frac{2}{3}\right)^{n-1}\]This formula allows us to find any term in the sequence effortlessly by substituting the term number for \(n\). It's a powerful tool providing insight into the sequence's progression.