91Ó°ÊÓ

In the realm of geometric sequences, the common ratio is pivotal for understanding the sequence's behavior. Put simply, it's the fixed number you multiply the previous term by to get to the next term. For instance, in the sequence from our exercise:

• The first term is 1000.
• The next term is 1500.

To find out how these two numbers are related, divide 1500 by 1000. You’ll find the common ratio, denoted by the letter \(r\), is 1.5.
This ratio tells us that each subsequent term is 1.5 times the prior one.
Verifying this is simple. If you continue dividing each term by its predecessor:

• 2250 divided by 1500
• 3375 divided by 2250

In every case, the ratio checks out as 1.5. Understanding the common ratio is like having the key to unlocking the sequence's progression.

Convergence is an important concept when talking about series and sequences. It describes a series whose terms approach a specific limit. For a geometric series, whether it converges mainly depends on the common ratio \(r\).
If \(|r| < 1\), meaning the absolute value of \(r\) is less than one, the sum of the infinite series can be calculated.
This is because the terms of the series get progressively smaller, eventually nearing zero.
But if \(|r| \geq 1\), the series diverges. This means it doesn’t add up to a single number.

• In our example, the common ratio is 1.5.
• This number is greater than 1, causing the series not to converge.

The concept of convergence helps decide whether a series has a finite sum, hinting toward the behavior over infinite extensions.

An infinite series expands its terms indefinitely, adding up an endless sequence. With geometric series, determining whether this sum is finite involves the common ratio \(r\).
This isn't just about adding numbers forever but understanding the long-term trend.
To find a sum, two conditions must align:

• The series must be geometric.
• The common ratio \(|r|\) must be less than 1.

Under these conditions, the sum of an infinite series \(S\) can be calculated using this formula:\[S = \frac{a}{1 - r}\]where \(a\) is the first term and \(r\) the common ratio.
In the exercise given, since \(r\) is 1.5, which is greater than 1, this series doesn't conform to a sum. Therefore, it goes on without a bound, a fundamental aspect of understanding infinite series in practical mathematics.