Problem 25 - Algebra and Trigonometry

Question

Find the first five terms of the sequence, and determine whether it is geometric. If is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=\ln \left(5^{n-1}\right) $$

Step-by-Step Solution

Answer

The sequence is not geometric.

Step 1: Find the First Term

To find the first term of the sequence, substitute \( n = 1 \) into the sequence formula: \[ a_1 = \ln(5^{1-1}) = \ln(5^0) = \ln(1) = 0. \] The first term is 0.

Step 2: Find the Second Term

For the second term, substitute \( n = 2 \) into the sequence formula: \[ a_2 = \ln(5^{2-1}) = \ln(5^1) = \ln(5). \] The second term is \( \ln(5) \).

Step 3: Find the Third Term

For the third term, substitute \( n = 3 \) into the sequence formula: \[ a_3 = \ln(5^{3-1}) = \ln(5^2) = \ln(25). \] The third term is \( \ln(25) \).

Step 4: Find the Fourth Term

For the fourth term, substitute \( n = 4 \) into the sequence formula: \[ a_4 = \ln(5^{4-1}) = \ln(5^3) = \ln(125). \] The fourth term is \( \ln(125) \).

Step 5: Find the Fifth Term

For the fifth term, substitute \( n = 5 \) into the sequence formula: \[ a_5 = \ln(5^{5-1}) = \ln(5^4) = \ln(625). \] The fifth term is \( \ln(625) \).

Step 6: Check if the Sequence is Geometric

A sequence is geometric if there is a constant ratio between consecutive terms. Calculate the ratio between terms:\[ \text{ratio} = \frac{a_2}{a_1} = \frac{\ln(5)}{0}. \]Since the first term is zero, dividing by zero is undefined, implying the sequence is not geometric. However, let's compute the difference between logs for farther confirmation:The sequence of logs generally indicates exponential growth, as we evaluate further...\[ \text{ratio} = \frac{\ln(5)}{\ln(1)}, \frac{\ln(25)}{\ln(5)}, \frac{\ln(125)}{\ln(25)}, \frac{\ln(625)}{\ln(125)}. \]Simplifying each:\[ \frac{\ln(5)}{\ln(1)} \text{ is undefined}, \frac{\ln(25)}{\ln(5)} = 2, \frac{\ln(125)}{\ln(25)} = 1.6667, \frac{\ln(625)}{\ln(125)} = 1.25. \]These ratios are not consistent, confirming non-geometric status.

Key Concepts

Logarithms

Logarithms are mathematical operations that help us explore exponential growth and decay. A logarithm asks the question: "To what power must a specific base be raised to obtain a certain number?" For example, if we have \( \log_b(x) = y\), this means that the base \( b\) raised to the power \( y\) equals \( x\) (\(b^y = x\)).
In the context of our exercise, we work with natural logarithms where the base is the mathematical constant \( e\) (approximately equal to 2.71828).

Key properties of logarithms often used in sequences include:

• \( \ln(a^b) = b \cdot \ln(a)\)
• \( \ln(1) = 0\)
• \( \ln(e) = 1\)

These properties simplify calculations when solving problems involving exponential expressions, such as our sequence formula \( a_n = \ln(5^{n-1} )\).
Understanding these logarithmic properties allows you to efficiently compute terms in a sequence, helping you recognize patterns and determine their nature.

Common Ratio

The common ratio is a factor that helps identify geometric sequences. In a geometric sequence, each term after the first is generated by multiplying the previous term by this constant factor. In a truly geometric sequence, this ratio remains constant for all consecutive term pairs.
Finding the common ratio involves:

• Taking any two consecutive terms, say \(a_2\) and \(a_1\), and computing \(\frac{a_2}{a_1}\).
• If consistent across all terms, this ratio confirms the presence of a geometric sequence.

For our exercise, we faced a challenge with the undefined transition from \(a_1 = 0\). Since dividing by zero doesn't yield a valid numerical result, the sequence is not considered geometric based on the common ratio definition. This initial undefined ratio interferes with establishing a steady multiplication pattern required in geometric sequences.

Sequence Formula

The sequence formula defines how to compute any term in a series based on its position. It offers a blueprint for generating terms without extensive calculations from scratch.

In our exercise, the sequence is defined by: \(a_n = \ln(5^{n-1} )\). This sequence uses natural logarithms to dictate term growth. Let's break this formula:

• \(5^{n-1}\) indicates that the base, 5, is raised to an exponentially increasing power as \(n\) steps forward.
• Applying the logarithm \(\ln\), translates the exponential expression into a log scale sequence to determine each term's value.

When analyzing such formulas, observe:

• How the power of the base changes with \(n\)
• The role of the logarithm in adjusting outputs

By understanding the sequence formula intricately, you can predict and conceptualize the behavior of sequences, identifying if they follow geometric, arithmetic, or other unique progressions.