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A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of sequence progresses by a constant factor, making it simple to predict future terms if the common ratio and initial term are known.
For example, consider the geometric sequence defined by the initial term, also known as the first term, and the common ratio. In mathematical terms, a geometric sequence can be represented as:

• \(a, ar, ar^2, ar^3, \ldots\)

Where \(a\) is the initial term, and \(r\) is the common ratio. Each term decreases or increases exponentially depending on whether the common ratio is a fraction or greater than one. In our exercise, the geometric sequence is based on \(\left(\frac{2}{5}\right)^2\), indicating a common ratio that involves repeating fractions, leading to a descending pattern of values.

The geometric sum formula allows us to find the sum of terms in a geometric sequence quickly. Instead of adding each term manually, which can be tedious especially for large numbers of terms, the sum formula offers a quick method. The formula for the sum \(S_n\) of the first \(n\) terms of a geometric sequence is:

• \(S_n = \frac{a(1-r^n)}{1-r}\)

Where \(a\) is the initial term, \(r\) is the common ratio, and \(n\) is the number of terms.
In the provided exercise, this formula is used to compute the sum of 21 terms of a geometric sequence. By inputting the specific values for \(a\), \(r\), and \(n\), the calculation becomes efficient and straightforward, yielding the result without error-prone manual addition.

The initial term of a geometric sequence is the first term from which the sequence begins. It is sometimes symbolized by the letter \(a\) and plays a crucial role in predicting subsequent terms and in the sum formula.
To determine the initial term from the sequence \(\left(\frac{2}{5}\right)^{2k}\),we calculate when \(k = 0\), simplifying to:

• \(a = \left(\frac{2}{5}\right)^0 = 1\)

This simplification tells us that the initial term here is 1, from which all subsequent terms are derived by multiplying by the common ratio. Understanding the initial term helps clarify how quickly terms will grow or shrink in the sequence.

The common ratio in a geometric sequence is the constant factor between successive terms. Represented by \(r\), this ratio determines whether the sequence grows, shrinks, or remains constant.
Finding the common ratio involves examining the given expression or terms in the sequence. In our exercise, each term is derived from multiplying the previous term by \(\left(\frac{2}{5}\right)^2\), which simplifies to a common ratio of \(\left(\frac{4}{25}\right)\).

• It's important to check:
  • If \(|r| < 1\), the sequence decreases.
  • If \(|r| > 1\), it increases.

In this case, the common ratio being \(\frac{4}{25}\) signifies a decreasing geometric progression, as each term is a fraction of the previous one.