Problem 32 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 5,5^{c+1}, 5^{2 c+1}, 5^{3 c+1}, \dots $$

Step-by-Step Solution

Answer

Common ratio: 5^c, Fifth term: 5^{4c+1}, nth term: 5^{cn-c+1}.

Step 1: Understand the First Term

Identify the first term of the sequence. The first term is given as 5, which can be written as \(5^1\).

Step 2: Identify the Second Term

The second term of the sequence is \(5^{c+1}\).

Step 3: Calculate the Common Ratio

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

Step 4: Apply the Formula for the Fifth Term

The fifth term of a geometric sequence can be found using the formula \(a_n = a_1 \times r^{n-1}\), where \(n\) is the term position. Substitute the known values: \(a_5 = 5^1 \times (5^c)^{4} = 5 \times 5^{4c} = 5^{4c+1}\).

Step 5: Derive the General Formula for the nth Term

The nth term can be found using the same formula as above: \(a_n = a_1 \times r^{n-1}\). Here, \(a_n = 5^1 \times (5^c)^{n-1} = 5^{c(n-1)+1}\). Simplifying, we get \(5^{cn-c+1}\).

Key Concepts

Common Ratio

A fundamental component of any geometric sequence is the **common ratio**. It's the factor you multiply by to get from one term to the next. To find the common ratio, divide the second term of the sequence by the first term.
In our exercise, the sequence begins with 5, and the next term is expressed as \( 5^{c+1} \).

• First Term: \( 5^1 \)
• Second Term: \( 5^{c+1} \)

So, the common ratio \( r \) is \( \frac{5^{c+1}}{5^1} \). Using the properties of exponents, this simplifies to \( 5^c \).
The common ratio tells us how the sequence grows, as each term is the previous term multiplied by \( 5^c \). Knowing this ratio is crucial for predicting future terms in the sequence.

nth Term Formula

The **nth term formula** provides a way to calculate any term in the sequence without listing all the terms before it. This shortcut is defined by the formula: \[ a_n = a_1 \times r^{n-1} \]Where:

• \( a_n \) is the nth term
• \( a_1 \) is the first term
• \( r \) is the common ratio
• \( n \) is the term number

In the given sequence, the first term \( a_1 \) is 5, and the common ratio \( r \) is \( 5^c \).
When we substitute these into the formula, we derive that the nth term \( a_n \) is:\[ a_n = 5^1 \times (5^c)^{n-1} = 5^{cn-c+1} \]This expression allows you to find any term directly, knowing the common ratio and the position of the term in the sequence.

Fifth Term

Calculating specific terms like the **fifth term** in a geometric sequence involves using the nth term formula we discussed. You first identify the position of the term you want, in this case, as the fifth term (\( n = 5 \)).
Here’s how it plays out:

• First Term (\( a_1 \)): 5
• Common Ratio (\( r \)): \( 5^c \)
• Term Position (\( n \)): 5

Substitute these into our formula to find the fifth term:\[ a_5 = 5 \times (5^c)^{4} = 5 \times 5^{4c} = 5^{4c+1} \]The calculated fifth term, \( 5^{4c+1} \), relies on the sequence's growth pattern encoded in the common ratio and reflects the compounding effect over five positions. Simply using this formula saves time and ensures you calculate the term accurately.