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Chapter 12: Problem 21

Write the first five terms of each geometric sequence. $$ a_{1}=2, r=3 $$

Short Answer

Expert verified
The first five terms are 2, 6, 18, 54, and 162.

Step by step solution

01

Identify the General Formula for a Geometric Sequence

The general formula for a geometric sequence is given by: \[ a_n = a_1 \times r^{(n-1)} \] where \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
02

First Term Calculation

Given \( a_1 = 2 \), the first term is simply \[ a_1 = 2 \]
03

Second Term Calculation

To find the second term, use the formula with \( n = 2 \): \[ a_2 = 2 \times 3^{(2-1)} = 2 \times 3 = 6 \]
04

Third Term Calculation

Use the formula with \( n = 3 \) to find the third term: \[ a_3 = 2 \times 3^{(3-1)} = 2 \times 3^2 = 2 \times 9 = 18 \]
05

Fourth Term Calculation

For the fourth term, use \( n = 4 \): \[ a_4 = 2 \times 3^{(4-1)} = 2 \times 3^3 = 2 \times 27 = 54 \]
06

Fifth Term Calculation

Finally, use the formula with \( n = 5 \) to find the fifth term: \[ a_5 = 2 \times 3^{(5-1)} = 2 \times 3^4 = 2 \times 81 = 162 \]

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Key Concepts

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

geometric sequence formula
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio. The general formula for a geometric sequence is: \[ a_n = a_1 \times r^{(n-1)} \] where:
  • \(a_n\) = the nth term of the sequence
  • \(a_1\) = the first term of the sequence
  • \(r\) = the common ratio
  • \(n\) = the term number.
This formula helps us to find any term in the sequence if we know the first term and the common ratio.
It's like a mathematical recipe that takes an input (term number) and gives an output (term value).
common ratio
The common ratio \( r \) is a key element in geometric sequences. It is the factor that each term is multiplied by to get the next term.

Identifying the Common Ratio:
  • Given a sequence, you can find the common ratio by dividing any term by the previous term.
  • For example, if the first term is 2 and the second term is 6, then \( r = \frac{6}{2} = 3 \).
  • In our example, the common ratio is 3, meaning each term is three times the previous term.
An important thing to remember is that the common ratio can be positive or negative, but it should never be zero. The common ratio tells us how the sequence grows or shrinks.
term calculation
Calculating Terms in a Geometric Sequence involves using the geometric sequence formula. Let's calculate the first five terms of the sequence with first term \(a_1=2\) and common ratio \(r=3\):

First Term:
  • The first term is given: \(a_1=2\).
Second Term:
  • Using the formula with \(n=2\): \(a_2=2 \times 3^{(2-1)} = 6\).
Third Term:
  • Using the formula with \(n=3\): \(a_3=2 \times 3^{(3-1)}=18\).
Fourth Term:
  • Using the formula with \(n=4\): \(a_4=2 \times 3^{(4-1)} = 54\).
Fifth Term:
  • Using the formula with \(n=5\): \(a_5=2 \times 3^{(5-1)} = 162\).
By following these steps, you can calculate any term in the geometric sequence easily. Each step flows logically, ensuring you find the correct term based on the formula.

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Most popular questions from this chapter

A fully wound yo-yo has a string 40 in. long. It is allowed to drop, and on its first rebound it returns to a height 15 in. lower than its original height. Assuming that this "rebound ratio" remains constant until the yo-yo comes to rest, how far does it travel on its third trip up the string?

Solve each problem. The total assets of mutual funds operating in the United States, in billions of dollars, for each year during the period 2004 through 2008 are shown in the table. What were the average assets per year during this period? \(\begin{array}{|c|c|}\hline \text { Year } & {\text { Assets (in billions of dollars) }} \\ {2004} & {8107} \\ {2005} & {8905} \\ {2006} & {10,397} \\\ {2007} & {12,000} \\ {2008} & {9601} \\ \hline\end{array}\)

If the given sequence is geometric, find the common ratio \(r\) If the sequence is not geometric, say so. See Example 1 . $$ 5,15,45,135, \dots $$

Find a general term for each geometric sequence. $$ -2, \frac{2}{3},-\frac{2}{9}, \ldots $$

Write out each series and evaluate it. $$ \sum_{i=2}^{6}(i+3)(i-4) $$
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