Understanding the Explicit Formula for a Geometric Sequence: Step-by-Step Explanation and Example

Explicit Formula for Geometric Sequence

The explicit formula for a geometric sequence is given by:

\[a_n = a \cdot r^{n-1}\]

where:
– \(a_n\) is the value of the \(n\)th term in the sequence
– \(a\) is the first term of the sequence
– \(r\) is the common ratio

To understand how this formula is derived, let’s break it down step by step

The explicit formula for a geometric sequence is given by:

\[a_n = a \cdot r^{n-1}\]

where:
– \(a_n\) is the value of the \(n\)th term in the sequence
– \(a\) is the first term of the sequence
– \(r\) is the common ratio

To understand how this formula is derived, let’s break it down step by step.

In a geometric sequence, each term is found by multiplying the previous term by a fixed number called the common ratio (\(r\)).

For example, consider the geometric sequence: 2, 6, 18, 54, …
To find the next term in the sequence, we multiply the previous term by 3 (the common ratio):
\(2 \times 3 = 6, 6 \times 3 = 18, 18 \times 3 = 54, \ldots\)

To find the explicit formula, we need to find a pattern that relates the position of each term \(n\) to the first term \(a\) and the common ratio \(r\).

Notice that the exponent of \(r\) in each term is one less than the position of the term. For example, in the given sequence:
– The first term (\(a_1\)) has no \(r\) exponent
– The second term (\(a_2\)) has \(r\) raised to the power of 1 (since it is the second term)
– The third term (\(a_3\)) has \(r\) raised to the power of 2 (since it is the third term)
This pattern holds for all terms in a geometric sequence.

Using this pattern, we can now construct the explicit formula:
\[a_n = a \cdot r^{n-1}\]

Let’s use the formula to find the fifth term in the given sequence:
\(a_5 = a \cdot r^{5-1}\)
Since the first term \(a\) is 2 and the common ratio \(r\) is 3, we can substitute these values into the formula:
\(a_5 = 2 \cdot 3^{5-1}\)
Simplifying this expression, we get:
\(a_5 = 2 \cdot 3^4\)
\(a_5 = 2 \cdot 81\)
\(a_5 = 162\)

Therefore, the fifth term of the sequence is 162.

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