Problem 13 - A Graphical Approach to College Algebra

Question

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$\frac{4}{5}, 2,5, \frac{25}{2}, \dots$$

Step-by-Step Solution

Answer

\(a_5 = 31.25\) and \(a_n = \frac{4}{5} \left(\frac{5}{2}\right)^{n-1}\).

Step 1: Identify the First Term

The first term of the geometric sequence is the first element in the given sequence. Here, the first term \(a_1\) is \(\frac{4}{5}\).

Step 2: Find the Common Ratio

A geometric sequence can be defined by its common ratio, \(r\). To find this, divide the second term by the first term. \[ r = \frac{2}{\frac{4}{5}} = \frac{2 \cdot 5}{4} = \frac{10}{4} = \frac{5}{2} \] Hence, the common ratio \(r\) is \(\frac{5}{2}\).

Step 3: Use the Geometric Sequence Formula

The general term \(a_n\) for a geometric sequence can be found using the formula: \[ a_n = a_1 \cdot r^{n-1} \] Substitute \(a_1 = \frac{4}{5}\) and \(r = \frac{5}{2}\) into the formula.

Step 4: Calculate \(a_5\)

To find \(a_5\), plug \(n = 5\) into the general term formula:\[ a_5 = \frac{4}{5} \cdot \left(\frac{5}{2}\right)^{4} \] Calculate \(\left(\frac{5}{2}\right)^4 = \frac{625}{16}\):\[ a_5 = \frac{4}{5} \cdot \frac{625}{16} = \frac{4 \cdot 625}{5 \cdot 16} = \frac{2500}{80} = 31.25 \] Thus, \(a_5 = 31.25\).

Step 5: Express \(a_n\) as a General Term

Write the formula for the general term \(a_n\) using the identified first term and common ratio:\[ a_n = \frac{4}{5} \cdot \left(\frac{5}{2}\right)^{n-1} \] This formula can be used to find any specific term in the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that determines how the sequence progresses from one term to the next. It's a consistent factor that you multiply by one term to get the next term. To find it, simply divide the second term by the first term of the sequence.
For example, in the sequence \( \frac{4}{5}, 2, 5, \frac{25}{2}, \dots \) the second term is 2, and the first term is \( \frac{4}{5} \). You would compute the common ratio \( r \) by performing the division:

• \( r = \frac{2}{\frac{4}{5}} = \frac{10}{4} = \frac{5}{2} \).

The common ratio, \( \frac{5}{2} \), tells us that each successive term is \( \frac{5}{2} \) times the previous term.
This uniform factor makes calculations easy and predictable, as it remains constant throughout the entire sequence.

General Term Formula

The general term formula for a geometric sequence allows us to find any term in the sequence without listing out all the preceding ones. It relies on the first term and the common ratio to establish a pattern.
The formula is expressed as:

• \( a_n = a_1 \cdot r^{n-1} \)

Here, \( a_n \) represents the \( n^{th} \) term, \( a_1 \) is the first term, and \( r \) is the common ratio.
Let's apply this formula using our known values from the earlier example:

• First term \( a_1 = \frac{4}{5} \)
• Common ratio \( r = \frac{5}{2} \).

The formula helps us derive any term in the sequence. It's particularly useful for finding terms far along the sequence without manual calculation of all the preceding terms.

Geometric Sequence Formula

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, \( r \). Using the geometric sequence formula, we can express terms compactly and calculate them efficiently.
The geometric sequence formula is helpful for problems that involve complex sequences because it understands the interplay between terms, the ratio, and the position within the sequence.

• For a sequence starting with \( a_1 = \frac{4}{5} \) and having a common ratio \( \frac{5}{2} \), the general expression for any term \( a_n \) becomes:
• \( a_n = \frac{4}{5} \cdot \left(\frac{5}{2}\right)^{n-1} \)

This formula equips you with the tools to solve straightforward solutions as well as intricate patterns, enhancing both understanding and examination skills.