Problem 22 - Beginning and Intermediate Algebra

Question

Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=\frac{3}{5}, r=2 ; a_{6}$$

Step-by-Step Solution

Answer

The general term of the geometric sequence is \(a_{m} = \frac{3}{5} \cdot 2^{m-1}\) and the sixth term of the sequence is \(a_{6} = \frac{96}{5}\).

Step 1: Write down the given information

We are given the first term (\(a_{1} = \frac{3}{5}\)) and the common ratio (\(r=2\)). We will use these values to find the general term of the geometric sequence.

Step 2: Find the general term

The general term of a geometric sequence is given by the formula: \[a_{m} = a_{1} \cdot r^{m-1}\] Plug in the given values (\(a_{1} = \frac{3}{5}\) and \(r=2\)) into the formula: \[a_{m} = \frac{3}{5} \cdot 2^{m-1}\] This is the general term for the geometric sequence.

Step 3: Find the sixth term (\(a_{6}\))

Now we need to find the sixth term of the sequence (\(a_{6}\)). To do this, plug in \(m = 6\) into the general term found in Step 2: \[a_{6} = \frac{3}{5} \cdot 2^{6-1}\] Calculate the result: \[a_{6} = \frac{3}{5} \cdot 2^5 = \frac{3}{5} \cdot 32 = \frac{96}{5}\]

Step 4: Write down the answer

The general term of the geometric sequence is: \[a_{m} = \frac{3}{5} \cdot 2^{m-1}\] The sixth term of the sequence is: \[a_{6} = \frac{96}{5}\]

Key Concepts

General Term Formula

In a geometric sequence, each term after the first is the product of the previous term and a constant ratio, known as the common ratio. The general term formula is a foundational concept that allows you to calculate any term in the sequence without listing all previous terms. This formula is given by:

• \( a_{m} = a_{1} \cdot r^{m-1} \)

In this formula:

• \( a_{m} \) is the term you want to find.
• \( a_{1} \) is the first term of the sequence.
• \( r \) represents the common ratio.
• \( m \) is the position of the term in the sequence.

For example, given \( a_{1} = \frac{3}{5} \) and \( r = 2 \), the formula for any term in this sequence becomes \( a_{m} = \frac{3}{5} \cdot 2^{m-1} \). Simply substitute different values for \( m \) to find specific terms. This ease of calculation is a great advantage when working with large sequences or high numbered terms.

Common Ratio

The common ratio is a critical component of a geometric sequence. It determines how the sequence progresses from one term to the next. Once you know the common ratio, you have the key to unraveling the sequence.The common ratio (\( r \)) is the factor by which we multiply each term to get the next term in the sequence:

• If \( r > 1 \), the sequence grows exponentially.
• If \( 0 < r < 1 \), the sequence decreases and eventually approaches zero.
• If \( r = 1 \), all terms are constant, so it isn’t a geometric progression in the traditional sense.
• If \( r < 0 \), the sequence alternates in sign with each step.

For example, in the sequence where \( a_{1} = \frac{3}{5} \) and \( r = 2 \), the common ratio tells us that each term is twice the previous one. Such knowledge is crucial to manipulating the sequence and predicting its behavior.

Sequence Terms

The terms in a geometric sequence can be easily identified and calculated thanks to the general term formula and the common ratio. Each term follows the same multiplicative pattern from its predecessor.With the formula \( a_{m} = a_{1} \cdot r^{m-1} \), every term is just a step away:

• First term (\( a_{1} \)): As given, \( \frac{3}{5} \).
• Second term (\( a_{2} \)): \( \frac{3}{5} \cdot 2 = \frac{6}{5} \).
• Third term (\( a_{3} \)): \( \frac{6}{5} \cdot 2 = \frac{12}{5} \).
• ... until any \( n^{th} \) term is reached, such as the sixth term \( a_{6} = \frac{96}{5} \).

Understanding the arrangement of these sequence terms allows easy navigation and estimation of the sequence’s progression. Such fluency is immensely helpful when tackling complex mathematical problems involving geometric sequences.