Question

If four numbers are in geometric progression, then their logarithms will be in (a) GP (b) AP (c) HP (d) AGP

Short answer

Question: The logarithms of four numbers in geometric progression (GP) form a(n) ______. a) HP b) AP c) GP d) AGP Answer: b) AP and d) AGP

Step-by-step solution

Review properties of geometric progression (GP)

A sequence of numbers is said to be in geometric progression (GP) if each term is a constant multiple of the previous term, except for the first term. In other words, there exists a constant 'r' (called the common ratio) such that: a, ar, ar^2, ar^3, ... where a is the first term, and r is the common ratio. Now, let's use this information to establish the relationship between the logarithms of the four numbers in GP.

Obtain logarithmic form of the GP

Let the four numbers in GP be given by a, ar, ar^2, and ar^3, where 'a' is the first term, and 'r' is the common ratio. To find the logarithmic form of these numbers, we'll take their logarithms (base 10): \(\log(a), \log(ar), \log(ar^2), \log(ar^3)\)

Apply logarithm properties to simplify expressions

Use the logarithm product rule to simplify the logarithmic expressions. The product rule states that \(\log(a * b) = \log(a) + \log(b)\). Applying this rule to the logarithms of our GP numbers, we get: \(\log(a), \log(a) + \log(r), \log(a) + 2 \cdot \log(r), \log(a) + 3 \cdot \log(r)\)

Determine if logarithms form a GP, AP, HP, or AGP

Now that we have simplified the expressions, we'll analyse the relationship between the logarithms to see if they are in a Geometric progression (GP), Arithmetic progression (AP), Harmonic progression (HP), or Arithmetic-Geometric progression (AGP). Geometric progression (GP): Each term is a constant multiple of the previous term. Our logarithmic terms have the form of: \(\log(a), \log(a) + \log(r), \log(a) + 2 \cdot \log(r), \log(a) + 3 \cdot \log(r)\) Here, we don't observe a constant multiple being applied to each term. Arithmetic progression (AP): The difference between consecutive terms is constant, which can be represented as: \(a', a' + d, a' + 2d, a' + 3d, ...\) Comparing this with our logarithmic terms: \(\log(a), \log(a) + \log(r), \log(a) + 2 \cdot \log(r), \log(a) + 3 \cdot \log(r)\) We can see a common difference between consecutive terms, namely, \(\log(r)\). Therefore, the logarithms of four numbers in GP form an Arithmetic Progression (AP). Harmonic progression (HP): A sequence of numbers is in harmonic progression if the reciprocals of the terms are in AP. Our logarithmic terms do not satisfy this condition. Arithmetic-Geometric progression (AGP): A sequence of numbers is in AGP if the logarithms of the terms form an AP. This condition has already been met, so the logarithms of our four numbers in GP are also in AGP.

Choose the correct option

Based on our analysis, the logarithms of the four numbers in geometric progression (GP) form an Arithmetic Progression (AP) and an Arithmetic-Geometric Progression (AGP). Hence, the correct options are: (b) AP and (d) AGP

Key concepts

Geometric Progression

A geometric progression (GP) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Suppose we start with the first term 'a'. Then, the sequence follows the pattern:

• First term: \(a\)
• Second term: \(ar\)
• Third term: \(ar^2\)
• Fourth term: \(ar^3\)

Here, 'r' is the common ratio. This sequence allows us to quickly determine a specific term if we know the first term and the common ratio.
One important property of geometric progressions is that the ratio of successive terms is always the same, which can be particularly useful in various mathematical and scientific applications.
Understanding this concept is fundamental to later derive the relationships between other types of progressions related to GPs, such as the arithmetic progression formed by the logarithms of terms in a GP.

Arithmetic Progression

Arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is always the same. This difference is known as the common difference 'd'. For example, in the sequence where each term increases by 'd', the sequence would be:

• First term: \(a'\)
• Second term: \(a' + d\)
• Third term: \(a' + 2d\)
• Fourth term: \(a' + 3d\)

In the case of the logarithms from a geometric progression, we find that they form an arithmetic progression with a common difference of \(\log(r)\).
This occurs because logarithms, utilizing their properties, transform the multiplication involved in geometric progressions into addition, which results in an arithmetic progression.
The simplicity of APs makes them easy to use in solving problems involving sequences and helps illustrate the interconnectedness of logarithms and basic mathematical series.

Logarithmic Properties

Logarithmic properties are mathematical rules that simplify expressions and calculations involving logarithms. When working with sequences like geometric progressions, these properties become incredibly useful. Let's explore a key property involved in this scenario: the product rule.

• **Product Rule**: The logarithm of a product is the sum of the logarithms: \(\log(a \cdot b) = \log(a) + \log(b)\).

In the context of the problem, we applied this product rule to terms like \(\log(ar^2)\), which expanded to \(\log(a) + 2 \cdot \log(r)\).
By understanding and using logarithmic properties, we can transform multiplicative sequences (geometric progressions) into additive sequences (arithmetic progressions), thus simplifying complex problems and making them more approachable.

Sequences and Series

Sequences and series are fundamental concepts in mathematics, dealing with ordered lists of numbers and their sums. Sequences can be finite or infinite, and they help in organizing numbers systematically. A few critical types include arithmetic and geometric sequences, each with unique characteristics and uses.

• **Sequence**: An ordered list of numbers (e.g., \(a, ar, ar^2, ar^3\) in geometric progression).
• **Series**: The sum of the terms of a sequence.

Understanding the relationship between sequences and series allows us to see how terms grow or decrease, and how summations appear naturally out of repetitive patterns.
In the context of our problem, the transformation of a geometric sequence into an arithmetic sequence (via logarithms) exemplifies the power of mathematical techniques to relate different forms and solve problems across contexts.