91Ó°ÊÓ

A geometric progression is known for its constant ratio between consecutive terms. This constant factor is called the common ratio, and it is essential in determining the nature of the sequence. To verify if a sequence is a geometric progression, we can find the ratio between consecutive terms.

In our example sequence: 1, -\(\frac{1}{2}\), \(\frac{1}{4}\), -\(\frac{1}{8}\), ... , we calculate:

• First ratio: \(\frac{-\frac{1}{2}}{1} = -\frac{1}{2}\)
• Second ratio: \(\frac{\frac{1}{4}}{-\frac{1}{2}} = -\frac{1}{2}\)
• Third ratio: \(\frac{-\frac{1}{8}}{\frac{1}{4}} = -\frac{1}{2}\)

These identical ratios confirm that the sequence is indeed a geometric progression, with a common ratio of \(-\frac{1}{2}\). Recognizing this common ratio helps us understand the growth or decay pattern in sequence of numbers.

To find a specific term in a geometric progression, we use a simple formula involving the common ratio. The formula is:

\[a_n = a_1r^{n-1}\]
where

• \(a_1\) represents the first term,
• \(r\) is the common ratio,
• \(n\) is the position of the term we want to find.

In this scenario, the first term \(a_1\) is 1, the common ratio \(r\) is \(-\frac{1}{2}\), and we need to find the seventh term, so \(n = 7\).

Plugging in these values:

\[a_7 = 1\cdot(-\frac{1}{2})^{7-1} = \frac{1}{64}\]
This calculation shows that the seventh term in this particular sequence is \(\frac{1}{64}\). This formula is handy because it allows us to directly find any term in the sequence without listing all preceding terms.

The sum of terms in a geometric progression provides valuable information about the overall pattern and trend of the sequence. For a finite number of terms, the sum is calculated using the formula:

\[S_n = \frac{a_1(r^n - 1)}{r - 1}\]
where

• \(a_1\) is the first term,
• \(r\) is the common ratio,
• \(n\) is the total number of terms we are summing.

In this example, the task is to find the sum of the first seven terms with \(a_1 = 1\), \(r = -\frac{1}{2}\), and \(n = 7\).

Substitute these into the formula:

\[S_7 = \frac{1((-\frac{1}{2})^7 - 1)}{-\frac{1}{2} - 1} = \frac{129}{192}\]
This computation reveals that the sum of the first seven terms of the series is \(\frac{129}{192}\). Knowing this sum helps in understanding the collective behavior of the terms, particularly useful in financial calculations and scientific measurements.