Problem 36 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 5,5^{c+1}, 5^{2 c+1}, 5^{3 c+1}, \ldots $$

Step-by-Step Solution

Answer

Common ratio: \(5^c\); Fifth term: \(5^{4c+1}\); nth term: \(5^{cn-c+1}\).

Step 1: Identify the First Term

The first term of the sequence is given directly from the provided sequence. The first term is the number that appears first in the sequence. Here, the first term \( a_1 \) of the sequence is \( 5 \).

Step 2: Calculate the Common Ratio

The common ratio \( r \) of a geometric sequence is found by dividing the second term by the first term. So, \( r = \frac{5^{c+1}}{5} = 5^c \). Thus, the common ratio is \( 5^c \).

Step 3: Find the Fifth Term

To find the fifth term of a geometric sequence, use the formula for the nth term of a geometric sequence: \( a_n = a_1 \cdot r^{n-1} \). For the fifth term \( a_5 \), substitute \( n = 5 \), \( a_1 = 5 \), and \( r = 5^c \). Then, \[ a_5 = 5 \cdot (5^c)^{4} = 5 \cdot 5^{4c} = 5^{4c+1}. \] Thus, the fifth term is \( 5^{4c+1} \).

Step 4: Determine the Formula for the nth Term

The nth term \( a_n \) of a geometric sequence is given by the formula \( a_n = a_1 \cdot r^{n-1} \). For this sequence, substitute \( a_1 = 5 \) and \( r = 5^c \). Thus, \[ a_n = 5 \cdot (5^c)^{n-1} = 5 \cdot 5^{c(n-1)} = 5^{cn-c+1}. \] Hence, the formula for the nth term is \( 5^{cn-c+1} \).

Key Concepts

Understanding the Common Ratio

To understand a geometric sequence, one of the most crucial elements to identify is the common ratio. This ratio, represented by "r", is the value you multiply by to get from one term to the next in the sequence. In the given sequence, after identifying the first term as 5, the second term is represented as \( 5^{c+1} \). The common ratio is the factor that links one term to the next. You can find "r" by dividing the second term by the first term:

• The second term is \( 5^{c+1} \)
• The first term is 5
• Thus, the common ratio \( r \) is \( \frac{5^{c+1}}{5} = 5^c \)

In this scenario, you can imagine the sequence evolves by multiplying each term by \( 5^c \), revealing the consistent pattern in a geometric sequence.

The Formula for the nth Term

Finding the formula for the nth term in a geometric sequence allows you to calculate any term without listing each one. The general formula for the nth term \( a_n \) is given by:

\[ a_n = a_1 \cdot r^{n-1} \]
Here, \( a_1 \) is the first term and "r" is the common ratio. In our sequence, we've identified:

• First term \( a_1 = 5 \)
• Common ratio \( r = 5^c \)

Substituting these into the formula, we get:
\[ a_n = 5 \cdot (5^c)^{n-1} = 5^{c(n-1)+1} = 5^{cn-c+1} \]

This formula is powerful as it provides a direct route to the nth term, working behind the scenes of the visible sequence.

Finding the Fifth Term

The fifth term, denoted \( a_5 \), can be found using the same nth term formula we've established. This utilizes both the first term and the common ratio to methodically find any position in the sequence.

Recall the formula:
\[ a_n = a_1 \cdot r^{n-1} \]

• Use \( a_1 = 5 \)
• Use \( r = 5^c \)
• Use \( n = 5 \)

Substituting these values, you get:
\[ a_5 = 5 \cdot (5^c)^{4} = 5 \cdot 5^{4c} = 5^{4c+1} \]

The result shows that the fifth term is \( 5^{4c+1} \). This highlights how the formula simplifies calculating any term in a geometric pattern without tracing through each previous term.