91Ó°ÊÓ

In the world of sequences and series, understanding individual series terms is essential. A series, in general, is a sum of terms of a sequence. Each term in the series can be expressed as a function of its position in the sequence.
For example, in the given series, the terms are distinct components of the sequence like:

• First term: 1
• Second term: \(2z\)
• Third term: \((2z)^2\)
• Fourth term: \((2z)^3\)

Identifying these individual terms allows us to analyze the pattern or rule that the series follows. This pattern is crucial to determine if a series is geometric, arithmetic, or neither.

The common ratio is a fundamental component in the characterization of a geometric series. It is the factor that you multiply by to get from one term to the next. It remains consistent throughout the series, which is the distinctive characteristic of a geometric series.
To find the common ratio in a geometric series, you divide any term by its previous term. Let's do this for our series:

• The ratio of the second term to the first term \(\frac{2z}{1} = 2z\)
• The ratio of the third term to the second term \(\frac{(2z)^2}{2z} = 2z\)
• The ratio of the fourth term to the third term \(\frac{(2z)^3}{(2z)^2} = 2z\)

Each ratio calculation gives us \(2z\), affirming the series is geometric with a common ratio of \(2z\). This consistency in the common ratio is what allows the series to be identified as a geometric series.

A geometric progression, often referred to as a geometric sequence, is when each term in the series is derived by multiplying the previous term by a fixed, non-zero number called the common ratio. This property makes geometric sequences incredibly predictable and easy to calculate if the first term and the common ratio are known.
The given series is an example of a geometric progression because:

• The first term \(a_1\) is 1.
• The consistent common ratio \(r\) throughout the series is \(2z\).

This pattern means each term is the product of the first term and a power of the common ratio, expressed as \(a_n = a_1 \cdot r^{(n-1)}\). Thus,

• The second term is \(1 \cdot (2z)^1 = 2z\)
• The third term is \(1 \cdot (2z)^2 = (2z)^2\)
• The fourth term is \(1 \cdot (2z)^3 = (2z)^3\)

Understanding this exponential relationship makes working with geometric progressions straightforward, enabling easy predictions and calculations of further terms in the sequence.