Problem 36 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern

Observe the given sequence: \(5, 5^{c+1}, 5^{2c+1}, 5^{3c+1}, \ldots\). Each term seems to follow an exponential pattern where the exponents are in arithmetic progression. This suggests a geometric sequence.

Step 2: Find the First Term and Common Ratio

The first term of the sequence (\(a_1\)) is \(5\). The second term is \(5^{c+1}\). To find the common ratio \(r\), divide the second term by the first term: \[ r = \frac{5^{c+1}}{5} = 5^c. \] Thus, the common ratio \(r\) is \(5^c\).

Step 3: Determine the Fifth Term

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

Step 4: Find the General nth Term Formula

The formula for the nth term \(a_n\) of a geometric sequence is \(a_1 \cdot r^{n-1}\). Substitute \(a_1 = 5\) and \(r = 5^c\): \[ a_n = 5 \cdot (5^c)^{n-1} = 5 \cdot 5^{c(n-1)} = 5^{cn+1-c}. \] Simplifying, the general formula for the nth term is \(5^{cn-c+1}\).

Key Concepts

Common Ratio

A geometric sequence is characterized by having a consistent ratio between each consecutive term, known as the "common ratio." Understanding this concept is essential, as the common ratio determines how the sequence progresses. For the sequence given in the exercise, the first term is 5, and the second term is \(5^{c+1}\).
To find the common ratio \(r\), divide the second term by the first:

• First term \(a_1\) is 5
• Second term is \(5^{c+1}\)
• Common ratio \(r = \frac{5^{c+1}}{5} = 5^c\)

Therefore, in this sequence, each term (after the first) is obtained by multiplying the previous term by \(5^c\). Understanding this ratio is crucial for predicting any subsequent terms and verifying that a sequence is truly geometric.

Exponential Pattern

Geometric sequences often exhibit a unique "exponential pattern." This term refers to the way in which the terms of the sequence can be expressed as powers of a base number. In our sequence, the base number is 5. The exponents of each term form an arithmetic progression. This means that the difference between consecutive exponents is constant.
Let's examine our sequence:

• First term: \(5 = 5^1\)
• Second term: \(5^{c+1}\)
• Third term: \(5^{2c+1}\)
• Fourth term: \(5^{3c+1}\)

The exponents here are 1, \(c+1\), \(2c+1\), \(3c+1\), which clearly increase by \(c\) each step. This arithmetic progression of exponents is what gives this sequence its exponential pattern. Recognizing such patterns can make it much easier to find any term within the sequence and understand its behavior as \(n\) becomes very large.

Nth Term Formula

To define any individual term in a geometric sequence, we use the "nth term formula." This formula gives us a mathematical expression to calculate any term in the sequence without needing to list all the previous terms.
For a geometric sequence, the nth term \(a_n\) can be calculated using:

• First term \(a_1\)
• Common ratio \(r\)

The formula is:\[ a_n = a_1 \cdot r^{n-1} \]Substituting the values from our specific sequence:

• \(a_1 = 5\)
• \(r = 5^c\)
• \(a_n = 5 \cdot (5^c)^{n-1} = 5^{cn-c+1}\)

This formula allows you to calculate any nth term by plugging in the value of \(n\). It simplifies the process of finding specific terms, like the 5th term, as calculated in the original solution. Being familiar with the nth term formula is a powerful tool for working with geometric sequences.