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Chapter 14: Problem 98

Explain how to find the general term of a geometric sequence.

Short Answer

Expert verified
The general term of a geometric sequence, with initial term (a) and common ratio (r), can be found using the formula \( a \cdot r^{(n - 1)} \)

Step by step solution

01

Identify Initial Term and Common Ratio

The first thing is to identify is the initial term (a), which is simply the first term in the sequence. The common ratio (r) is found by dividing any term in the sequence by the preceding term.
02

Express the General Term

The next thing is to express the general term using the formula. Each term can be found from the previous one by multiplying by the common ratio, thus the nth term can be found by starting with the first term (a) and multiplying by the common ratio (r) a certain number of times (n-1 times to be precise). Thus, the general term, often represented as \( T_n \), can be found using this formula: \( T_n = a \cdot r^{(n - 1)} \)
03

Provide an Example

Let's provide an example, if you have a geometric sequence that starts with 2 and has a common ratio of 3 (e.g. 2, 6, 18, 54 ...). The general term of the sequence, \( T_n \), is given by: \( T_n = 2 \cdot 3^{(n - 1)} \)

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