91Ó°ÊÓ

In a geometric sequence, the general term refers to a formula that allows us to find any term in the sequence based on its position. Understanding the general term is crucial because it provides a way to calculate sequence terms quickly without listing all preceding terms.
The general term formula for a geometric sequence is given by:

• \( a_n = a_1 \cdot r^{n-1} \)

Here, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. This formula defines a pattern that relates any term in the sequence to its position number \( n \).
In our example, the general term for the sequence is \( a_m = (-1) \cdot 3^{m-1} \). This formula derives from knowing the first term of the sequence \( a_1 = -1 \) and the common ratio \( r = 3 \). By substituting these values into the general term formula, we create a specific expression that describes our sequence.

The common ratio in a geometric sequence is the factor by which we multiply a term to get the next term. It is a key component of geometric sequences, distinguishing them from other types of sequences.
The common ratio \( r \) can be found by dividing any term by the previous term (assuming all terms are non-zero). Its role in the general term formula \( a_n = a_1 \cdot r^{n-1} \) is to scale the growth or decay of the sequence depending on its value:

• If \( r > 1 \), the sequence grows (each term is larger than the last).
• If \( 0 < r < 1 \), the sequence shrinks.
• If \( r < 0 \), the sequence alternates in sign.

In the example provided, the common ratio is \( r = 3 \). This means each term is three times the value of the previous one. Knowing the common ratio allows us to understand the sequence’s progression pattern and to use it in the nth term formula.

The nth term formula is an essential tool in understanding geometric sequences. It provides a way to calculate any term in the sequence without needing to compute all preceding terms when the general term is established.
For a geometric sequence, the nth term formula is expressed as:

• \( a_n = a_1 \cdot r^{n-1} \)

Where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of the term you wish to find. This formula is integral because it encapsulates the entire behavior of the sequence using only a position \( n \), its starting value \( a_1 \), and a repeated multiplier \( r \).
In the solution example, this formula allows us to find the general and specific terms efficiently. By knowing \( a_1 = -1 \) and \( r = 3 \), and plugging them into the nth term formula, we can derive the expression \( a_m = (-1) \cdot 3^{m-1} \). This clear and straightforward calculation is applicable to any term \( m \).

Sequence term calculation is the act of finding any specific term within a sequence using the established general term formula. It is a practical application of understanding the sequence’s growth or decay pattern.
To calculate a particular term, substitute the desired term number into the general term's formula. This replacement isolates the specific term, enabling precise computation:

• \( a_m = a_1 \cdot r^{m-1} \)

Step-by-step substitution and simplification follow this formula to yield a specific term. For example, to find the fifth term \( a_5 \) in our problem, we use:

• Substitute \( m = 5 \) into the formula \( a_m = (-1) \cdot 3^{m-1} \)
• Compute \( a_5 = (-1) \cdot 3^{5-1} = (-1) \cdot 3^{4} = (-1) \cdot 81 = -81 \)

Understanding sequence term calculations is crucial as it allows one to navigate directly to any point in the sequence with accuracy and ease.