Problem 88 - Precalculus Mathematics 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 $$\log a_{1}, \log a_{2}, \log a_{3}, \dots$$ is an arithmetic sequence, and find the common difference.

Step-by-Step Solution

Answer

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

Step 1: Understand the Definition of a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant non-zero number called the common ratio \(r\). Given \(a_1, a_2, a_3, \ldots\), the terms can be expressed as \(a_2 = a_1 \cdot r\), \(a_3 = a_2 \cdot r = a_1 \cdot r^2\), and so forth. In general, the \(n\)-th term \(a_n\) is given by \(a_n = a_1 \cdot r^{n-1}\).

Step 2: Convert the Geometric Sequence to Logarithms

We are given the sequence \(\log a_1, \log a_2, \log a_3, \ldots\). Using the expression for the \(n\)-th term of the sequence \(a_n = a_1 \cdot r^{n-1}\), the logarithms of the terms are: \(\log a_n = \log(a_1 \cdot r^{n-1}) = \log a_1 + \log r^{n-1}.\)

Step 3: Simplify the Logarithmic Expressions

Apply the logarithmic identity \(\log(xy) = \log x + \log y\) to the expression from Step 2: \(\log a_n = \log a_1 + (n-1) \log r.\)

Step 4: Identify the Arithmetic Sequence

Notice that the expression \(\log a_1 + (n-1) \log r\) is in the form of an arithmetic sequence, where the first term corresponds to \(\log a_1\) and the common difference is \(\log r\). Therefore, the sequence \(\log a_1, \log a_2, \log a_3, \ldots\) is indeed an arithmetic sequence with a common difference \(\log r\).

Step 5: Conclusion

The sequence of logarithms \(\log a_1, \log a_2, \log a_3, \ldots\) forms an arithmetic sequence. The common difference of this arithmetic sequence is \(\log r\).

Key Concepts

Logarithms

Logarithms are an essential concept in mathematics, particularly when dealing with sequences and exponential relationships. A logarithm can be thought of as the opposite of exponentiation. It answers the question, "To what power must we raise a certain base to obtain a particular number?" If you are given an expression like \(\log_b(x)\), it is asking you to find the power, say \(y\), where \(b^y = x\).

• Logarithm of a Product: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• Logarithm of a Power: \(\log_b(x^k) = k\log_b(x)\)
• Logarithm of a Quotient: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)

Logarithms are very handy for transforming complex multiplicative sequences into simpler additive sequences, as seen in the transformation of a geometric sequence into an arithmetic sequence.

Arithmetic Sequence

An arithmetic sequence is a list of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference and remains unchanged throughout the sequence. For example, in the sequence 2, 4, 6, 8, 10, the common difference is 2, because every term increases by 2 as you move from one to the next.
The general form of an arithmetic sequence can be expressed as follows:

• First Term: \(a_1\)
• Common Difference: \(d\)
• \(n\)-th Term: \(a_n = a_1 + (n-1) \cdot d\)

In the exercise we discussed, the sequence of logarithms \(\log a_1, \log a_2, \log a_3, \ldots\) is an arithmetic sequence with a common difference, as shown through the transformation of the geometric sequence.

Common Ratio

The common ratio is a key characteristic of a geometric sequence. It defines the factor by which each term in the sequence is multiplied to obtain the subsequent term. For a sequence to be geometric, this ratio must remain constant.
Consider the example of the geometric sequence 3, 6, 12, 24, where the common ratio is calculated by dividing any term by its preceding term. This gives \(\frac{6}{3} = \frac{12}{6} = \frac{24}{12} = 2\).
In the case of the initial exercise, the common ratio \(r\) is greater than zero, which is crucial since it affects the calculation of the terms \(a_n\) and consequently the transformation to the arithmetic sequence of logarithms. The common ratio's logarithm becomes the common difference in the arithmetic sequence formed by the logarithms of terms of the geometric sequence.

Common Difference

The concept of a common difference is central to understanding arithmetic sequences. As mentioned previously, it represents the consistent amount added (or subtracted) between consecutive terms in an arithmetic sequence.
In the context of the original exercise, transforming a geometric sequence into an arithmetic sequence through the use of logarithms allows us to determine the common difference of the new sequence. Here, this common difference is \(\log r\), where \(r\) is the common ratio of the initial geometric sequence.

• This transformation is possible due to the logarithmic property: \(\log(a \cdot b) = \log(a) + \log(b)\)
• Thus, applying the logarithm across terms, we obtain: \(\log a_n = \log a_1 + (n-1) \log r\), showing "\(\log r\)" as the common difference.

Understanding the relationship between common difference and common ratio via logarithms showcases the interplay of arithmetic and geometric sequences in mathematics.