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Chapter 11: Problem 31

Find the nth, or general, term for each geometric sequence. $$1,-1,1,-1, \ldots$$

Short Answer

Expert verified
\(a_n = (-1)^{n-1}\)

Step by step solution

01

Identify the first term

The first term of the geometric sequence is denoted as \(a\). For this sequence, \(a = 1\).
02

Find the common ratio

The common ratio \(r\) can be found by dividing the second term by the first term. Here, \(r = \frac{-1}{1} = -1\).
03

Use the formula for the nth term

The formula for the nth term of a geometric sequence is \(a_n = a \, r^{n-1}\).
04

Substitute the values

Substitute \(a = 1\) and \(r = -1\) into the formula: \(a_n = 1 \cdot (-1)^{n-1}\).

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Key Concepts

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

nth term
The nth term of a geometric sequence represents a specific term's position in the sequence. In simpler terms, it tells us the value of any term in the series when we know its position number (n). For example, in the sequence, the first term can be written as \(a\), the second term as \(a_2\), and so on. In our given example, the formula for the nth term is:
    \(a_n = 1 \cdot (-1)^{n-1}\). That helps us quickly find any term without writing out the whole sequence.
This makes working with long or complex sequences much easier!
common ratio
The common ratio is a crucial part of understanding geometric sequences. It tells us how we get from one term to the next by multiplying by the same number every time. To find it, we divide one term by the term before it. In our sequence, the second term is -1 and the first term is 1, so:
    \(r = \frac{-1}{1} = -1\)
Knowing the common ratio helps us understand the sequence's pattern and predict future terms.
formula for geometric sequence
The formula for finding the nth term of a geometric sequence is extremely powerful. For any geometric sequence, it's written as:
    \[a_n = a \cdot r^{n-1}\]
Here, \(a\) is the first term and \(r\) is the common ratio. In our example:
    \[a_n = 1 \cdot (-1)^{n-1}\]
This formula allows us to calculate the value of any term in the sequence without needing to list all terms before it. It鈥檚 a fundamental tool for working with geometric sequences efficiently.

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