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The first term in a sequence—or a series—is crucial because it sets the stage for all that follows. In a geometric series, this term is represented by the variable \(a\). It is simply the initial number from which the series starts. For the infinite geometric series provided, the first term is 20.

Identifying the first term is straightforward: look at the very first number in the sequence. This number is essential as it not only defines the start but is also a part of the formula used to determine the sum of the series.

• This term serves as the baseline to establish the growth and direction of the series.
• It's the starting point to which the common ratio is applied repeatedly.

Understanding this key concept helps in immediately grasping how the series begins and lays the foundation for further calculations.

The common ratio in a geometric series is the constant factor between consecutive terms. It is represented by the variable \(r\). To find this ratio, you take any term and divide it by the previous term. For the given series, if you divide the second term \(-1\) by the first term \(20\), you get the common ratio \(-0.05\).

Moreover, the common ratio provides information on how the series behaves:

• If \(r > 1\), the terms increase exponentially.
• If \(0 < r < 1\), the series decreases but converges towards a sum.
• If \(r < 0\), the series terms alternate in sign, bouncing between positive and negative values.

For this particular series, having a common ratio of \(-0.05\) signifies that the series will oscillate as it proceeds.

An infinite geometric series has a fascinating property: it can have a finite sum when the common ratio \(|r| < 1\). The formula to calculate the sum \(S\) of an infinite geometric series is:\[ S = \frac{a}{1 - r} \]This is provided that the absolute value of the common ratio is less than one, ensuring that the series converges.

Here's a quick breakdown:

• \(a\) is the first term of the series.
• \(r\) is the common ratio.

To find the sum of the series starting with a first term \(20\) and a common ratio of \(-0.05\), substitute these values into the formula:\[ S = \frac{20}{1 - (-0.05)} = \frac{20}{1.05} \]Evaluating this expression gives \(S \approx 19.05\), highlighting the power and utility of using this simple yet profound formula to ascertain the sum of an otherwise infinite series.