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Chapter 12: Problem 12

Find a general term for each geometric sequence. $$ -3, \frac{3}{2},-\frac{3}{4}, \ldots $$

Short Answer

Expert verified
\[ a_n = -3 \cdot \left( -\frac{1}{2} \right)^{n-1} \]

Step by step solution

01

Identify the First Term

The first term of the geometric sequence is given as onumber a_1 = -3.
02

Find the Common Ratio

To find the common ratio (onumber r), divide the second term by the first term. \[ r = \frac{\frac{3}{2}}{-3} = -\frac{1}{2} \]
03

Write the General Term Formula

The general term of a geometric sequence can be written as \( a_n = a_1 \cdot r^{n-1} \).
04

Substitute the Values

Substitute the values of the first term (onumber a_1 = -3) and the common ratio (onumber r = -\frac{1}{2}) into the general formula. \[ a_n = -3 \cdot \left( -\frac{1}{2} \right)^{n-1} \]

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Key Concepts

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

General Term Formula
The general term formula for a geometric sequence helps you find any term in the sequence without having to list all the terms. This is especially useful for sequences with many terms or large indices. The general term is given by the formula: $$ a_n = a_1 \times r^{(n-1)} $$ Here,
  • is the first term of the sequence
  • is the common ratio between the terms
  • is the position of the term you want to find
For our exercise, the first term is -3 and the common ratio is -1/2. Using the general term formula, we substitute these values to find any term in the sequence: $$ a_n = -3 \times \bigl( -1/2 \bigr)^{(n-1)} $$.
Common Ratio
The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term. You can find the common ratio by dividing any term by the previous term. In our exercise:
  • The first term is = -3
  • The second term is \frac{3}{2}
To find the common ratio , divide the second term by the first term: $$ r = \frac{\frac{3}{2}}{-3} = -\frac{1}{2} $$ This means each term is multiplied by -1/2 to get the next term. Since the common ratio is less than 1, the terms will get smaller in absolute value as you move along the sequence.
First Term
The first term of a geometric sequence is essential because it sets the starting point for the entire sequence. Without the first term, you cannot generate subsequent terms using the common ratio. In our exercise, we are given the first term:
This first term is crucial for the general term formula, where it is represented as . To find any term in the sequence using the general formula, you need to use: = -3 and multiply it by the common ratio raised to the power of one less than the term position, .

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