Problem 80 - Precalculus: Mathematical for Calculus

Question

If \(a_{1}, a_{2}, a_{3}, \ldots\) is a geometric sequence with a common ratio \(r>0\) and \(a_{1}>0,\) show that the sequence $$\log a_{1}, \log a_{2}, \log a_{3}, \ldots$$ is an arithmetic sequence, and find the common difference.

Step-by-Step Solution

Answer

The log sequence is arithmetic with a common difference \(\log r\).

Step 1: Express terms of a geometric sequence

Recall that in a geometric sequence, each term is the product of the previous term and the common ratio \(r\). So, the \(n^{th}\) term \(a_n\) of the sequence can be written as: \[ a_n = a_1 \cdot r^{n-1} \] where \(a_1\) is the first term. The sequence terms are therefore \(a_1, a_1 \cdot r, a_1 \cdot r^2, \ldots \).

Step 2: Apply logarithm to each term

We now take the logarithm of each term in the sequence. The sequence then transforms to: \( \log a_1, \log (a_1 \cdot r), \log (a_1 \cdot r^2), \ldots \). Using the property \(\log(ab)=\log a + \log b\), each term becomes: \[ \log a_n = \log a_1 + \log(r^{n-1}) = \log a_1 + (n-1)\log r \]

Step 3: Identify the sequence form

Observe that the logarithmic sequence you obtain is of the form: \[ b_n = \log a_1 + (n-1)\log r \] This resembles the form of an arithmetic sequence \(b_n = b_1 + (n-1)d\), where \(b_1 = \log a_1\) and \(d = \log r\).

Step 4: Conclude with the common difference

Since the transformed sequence \(\log a_1, \log a_2, \log a_3, \ldots \) fits the pattern of an arithmetic sequence, the common difference \(d\) between consecutive terms is given by \(\log r\). This confirms that the logarithmic sequence derived from a geometric sequence is indeed arithmetic.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the "common difference". In simpler terms, if you subtract any term from the next term in the sequence, you'll get the same result each time. This is why arithmetic sequences are straightforward and predictable.

For instance, in the sequence 3, 7, 11, 15, the difference between each pair of consecutive terms is 4. Therefore, the common difference is 4. If you start with a first term, say \(b_1\), then an arithmetic sequence can generally be represented as:

• \( b_n = b_1 + (n-1)d \)

where \(d\) is the common difference, and \(n\) is the position of the term in the sequence.

This concept is crucial in the given exercise, where a geometric sequence with logarithms applied becomes arithmetic.

Common Ratio

In a geometric sequence, each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the "common ratio." This ratio is a fundamental characteristic of geometric sequences, ensuring that although the numbers may grow rapidly or shrink, they do so consistently.

For example, in the sequence 5, 10, 20, 40, the common ratio \(r\) is 2, because each term is obtained by multiplying the previous term by 2.

Mathematically, you can express the \(n^{th}\) term of a geometric sequence as:

• \( a_n = a_1 \, r^{n-1} \)

with \(a_1\) as the first term and \(r\) as the common ratio. In the exercise, while the original sequence is geometric, taking the logarithm of each term transforms it into an arithmetic sequence. The logarithm effectively linearizes the multiplicative structure of the geometric sequence.

Logarithms

Logarithms are the mathematical inverses of exponents, and they're essential for dealing with rapid growth. They "compress" large scales down to more manageable sizes, which is why they're used in a variety of scientific fields.

Key properties of logarithms include:

• \( \log(ab) = \log a + \log b \)
• \( \log(a^b) = b \log a \)

These properties allow us to transform expressions and simplify calculations. For instance, in our context, when you take the logarithm of a geometric sequence, you turn multiplications into additions, effectively transforming the sequence into an arithmetic sequence.

In the provided exercise, by applying logarithms to the terms of a geometric sequence, the multiplicative relationship is transformed into an additive one, allowing it to be recognized and handled as an arithmetic sequence.

Common Difference

The common difference in an arithmetic sequence is the key to its linearity. It determines the "gap" between consecutive terms. In the transition from a geometric sequence to an arithmetic one through logarithms, understanding the common difference becomes crucial.

When you take the logarithm of each term in a geometric sequence, the repetitive multiplication by the common ratio \(r\) is turned into repetitive addition by \(\log r\). Thus, in our sequence example, the common difference \(d\) is exactly \(\log r\).

In a formula, the terms of an arithmetic sequence using the common difference can be written as:

• \( b_n = b_1 + (n-1) \cdot d \)

Here, \(d\) is the common difference, illustrating the linear increase (or decrease) from one term to the next. Recognizing this transformation helps us understand how different sequences relate under logarithms.