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Chapter 9: Problem 2

The \(n\)th term of a geometric sequence has the form \(a_{n}=\) ______.

Short Answer

Expert verified
The nth term of a geometric sequence \(a_n\) can be expressed as \(a_n = a_{1} \cdot r^{(n-1)\)}, where \(a_{1}\) is the first term and \(r\) is the common ratio.

Step by step solution

01

Identify key components of a geometric sequence

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant. This constant is known as the 'common ratio'. If we denote the first term by \(a_1\) and the common ratio by \(r\), then the second term \(a_2\) is given by \(a_1 \cdot r\), the third term \(a_3\) by \(a_2 \cdot r = a_1 \cdot r^2\), and so on. This pattern continues for all terms in the sequence.
02

Formulate the nth term of a geometric sequence

Following the pattern established from Step 1, we can extrapolate to determine the nth term of a geometric sequence, denoted by \(a_n\). This nth term is found by multiplying the first term, \(a_1\), by the common ratio, \(r\), raised to the power of (n-1). Therefore, we can express all terms in the geometric sequence using the formula: \(a_n = a_{1} \cdot r^{(n-1)}\).
03

Final conclusion

Hence, the nth term of a geometric sequence can be found using the formula \(a_n = a_1 \cdot r^{(n-1)}\). It's important to remember this formula when dealing with geometric sequences as it allows us to find any term in the sequence without knowing the previous one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

nth term
The concept of the nth term in a geometric sequence is crucial for finding any term in the sequence. A geometric sequence relies on having a first term and a common multiplier, known as the common ratio. Each term builds on the previous one, following a specific pattern. To find the nth term, you don't need to know all preceding terms; just the first term and the common ratio.
Imagine you have a sequence starting with 2, and the common ratio is 3. This means each term is 3 times the term before it. Starting with 2, the sequence becomes 2, 6, 18, and so on.
The formula for the nth term is derived by knowing that each subsequent term is the product of the previous term and the common ratio. So, the nth term formula becomes:
  • \( a_n = a_1 \cdot r^{(n-1)} \)
common ratio
In any geometric sequence, the common ratio is a fundamental concept that determines how the sequence progresses. The common ratio, often denoted as \( r \), is the number you repeatedly multiply with each term to get the next term. Knowing the common ratio is key to understanding the pattern of the sequence.
To find the common ratio, you simply divide any term by its preceding term. For example, in the sequence 2, 6, 18, if you divide 6 by 2, you get 3, and if you divide 18 by 6, you again get 3. Thus, 3 is the common ratio.
This common ratio allows us to easily calculate any term in the sequence, as shown in the nth term formula:
  • \( a_n = a_1 \cdot r^{(n-1)} \)
sequence formula
The sequence formula for a geometric sequence integrates both the first term and the common ratio to provide a means of calculating any desired term. The formula simplifies the process of finding terms, especially when dealing with terms that are far along in the sequence.
The sequence formula is expressed as \( a_n = a_1 \cdot r^{(n-1)} \). This formula is derived from observing the multiplication pattern inherent in geometric sequences.
Here's how it breaks down:
  • \( a_1 \) is the first term, which sets the sequence's starting point.
  • \( r \) is the common ratio, indicating how each term is generated from its predecessor.
  • \( n-1 \) in the exponent denotes how many times the common ratio is applied starting from the first term.
By using this formula, any number in the sequence can be calculated without needing to list out all preceding terms, saving time and effort, particularly for larger sequences.

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