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In a geometric sequence, each term is derived by multiplying the previous term by a fixed number. This fixed number is known as the **common ratio**. The common ratio, often denoted as \( r \), plays a vital role in determining the pattern of the sequence. For example, if the first term of a sequence \( a_1 \) is 1, and the common ratio is \( \frac{1}{2} \), then each subsequent term is found by multiplying the term before it by \( \frac{1}{2} \).

To find the common ratio of a sequence when it's not given, you divide the second term by the first term \( r = \frac{a_2}{a_1} \). This process can be repeated using any two consecutive terms in the sequence. Understanding the common ratio helps you predict and calculate terms in a geometric sequence with ease.

• A common ratio greater than 1 will cause the sequence to grow.
• A common ratio between 0 and 1 will cause the sequence to shrink.

The **series** of a sequence involves summing all the terms of the sequence up to a certain point. It's a cumulative total made from adding terms of a sequence together. In the context of a geometric sequence, the terms are added to create a geometric series.

For a finite geometric series, where you only have a specific number of terms, the sum \( S_n \) can be calculated with the formula:\[S_n = a_1 \left( \frac{1-r^n}{1-r} \right)\]where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

• The convergence of a series depends on the common ratio.
• If \( |r| < 1 \), the series will converge if continued infinitely.

Geometric series are widely used in areas such as finance and computer science, often found in interest calculation and algorithm design.

**Exponential decay** is a process where a quantity decreases at a rate proportional to its current value. In the context of geometric sequences, exponential decay describes how each term is smaller than the previous one, especially when the common ratio \( r \) is between 0 and 1.

This concept is very common in real-world applications like radioactive decay and depreciation of assets. When a sequence demonstrates exponential decay, it means the absolute value of each subsequent term is a fraction of the previous term.

• When \( 0 < r < 1 \), we're dealing with exponential decay.
• The smaller the ratio, the faster the decay occurs.

Understanding exponential decay helps in predicting how fast a quantity will approach zero, such as calculating the long-term value of investments or drugs in medicine.