Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for each geometric sequence. \(a_{n}=\left\\{-2, \frac{4}{3},-\frac{8}{9}, \frac{16}{27}, \ldots\right\\}\)

Step-by-Step Solution

Answer

The recursive formula is \(a_1 = -2\) and \(a_{n} = -\frac{2}{3} \, \cdot \, a_{n-1}\) for \(n > 1\).

Step 1: Identify the first term

The first term of the sequence \(a_1\) is the initial value of the sequence. Looking at the sequence given, the first term \(a_1 = -2\).

Step 2: Find the common ratio

To find the common ratio \(r\), divide the second term by the first term. So, \(r = \frac{\frac{4}{3}}{-2} = -\frac{2}{3}\). This is verified by checking the other terms: \(-\frac{8}{9} \div \frac{4}{3} = -\frac{2}{3}\) and \(\frac{16}{27} \div -\frac{8}{9} = -\frac{2}{3}\).

Step 3: Write the recursive formula

The recursive formula for a geometric sequence is given by \(a_1 = \text{first term}\) and \(a_{n} = r \cdot a_{n-1}\) for \(n > 1\). Plugging in our values, we get: \(a_1 = -2\) and \(a_{n} = -\frac{2}{3} \cdot a_{n-1}\).

Key Concepts

Recursive Formula

A recursive formula for a geometric sequence helps us find any term using its preceding term. The formula typically involves the first term and a common ratio. For this exercise, the recursive formula is structured as follows:

• The first term: \( a_1 = -2 \)
• The rule for the following terms: \( a_n = r \cdot a_{n-1}\)

Our specific sequence uses \( r = -\frac{2}{3} \) as the common ratio. Hence, the recursive formula for this sequence is \( a_n = -\frac{2}{3} \cdot a_{n-1} \) for \( n > 1 \). This approach allows us to efficiently calculate any term in the sequence by simply knowing the previous term.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next one. It's found by dividing any term by its preceding term. In our sequence:

• The second term is \(\frac{4}{3}\)
• The first term is \(-2\)

To find the common ratio \(r\), we calculate:\[r = \frac{\frac{4}{3}}{-2} = -\frac{2}{3}\]Thus, the common ratio \(-\frac{2}{3}\) is verified by checking subsequent term divisions, ensuring consistency across the whole sequence.

First Term

Identifying the first term of a sequence is crucial as it sets up the base for the recursive formula. In any geometric sequence, the first term \( a_1 \) serves as the starting point. Here, we have:

• First term (\(a_1\)): \(-2\)

This initial value is given explicitly in the sequence. Knowing this term allows us to determine the subsequent terms when combined with the common ratio of the sequence. Consequently, the whole sequence can be developed with just these key pieces.

Sequence Transformation

Sequence transformation involves understanding how each element of a sequence relates to the next. In geometric sequences, transformation is achieved by repeatedly applying the common ratio to the terms:

• Start with the first term: \(a_1 = -2\)
• Multiply by the common ratio: \(-\frac{2}{3}\)
• This generates the next term in the sequence.

Through this method, each term is a transformation from the former, following a consistent mathematical rule. This concept helps in predicting future terms or examining the progression of terms in the sequence.