Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 4
Find the first four terms of each geometric sequence. What is the common ratio? $$a_{n}=2 \cdot(3)^{n-1}$$
Short Answer
Expert verified
The first four terms are 2, 6, 18, and 54. The common ratio is 3.
Step by step solution
01
- Identify the General Formula
The general formula given for the geometric sequence is \( a_{n} = 2 \cdot (3)^{n-1} \).
02
- Calculate the First Term
To find the first term (\( a_1 \)), set \( n = 1 \): \[ a_1 = 2 \cdot (3)^{1-1} = 2 \cdot (3)^0 = 2 \cdot 1 = 2 \]
03
- Calculate the Second Term
To find the second term (\( a_2 \)), set \( n = 2 \): \[ a_2 = 2 \cdot (3)^{2-1} = 2 \cdot (3)^1 = 2 \cdot 3 = 6 \]
04
- Calculate the Third Term
To find the third term (\( a_3 \)), set \( n = 3 \): \[ a_3 = 2 \cdot (3)^{3-1} = 2 \cdot (3)^2 = 2 \cdot 9 = 18 \]
05
- Calculate the Fourth Term
To find the fourth term (\( a_4 \)), set \( n = 4 \): \[ a_4 = 2 \cdot (3)^{4-1} = 2 \cdot (3)^3 = 2 \cdot 27 = 54 \]
06
- Find the Common Ratio
The common ratio (\( r \)) can be found by dividing any term by the previous term. Using \( a_2 \) and \( a_1 \): \[ r = \frac{a_2}{a_1} = \frac{6}{2} = 3 \]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
common ratio
In a geometric sequence, each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. In our example, the geometric sequence is defined by the formula \(a_n = 2 \times (3)^{n-1}\). To find the common ratio (\r\text), we divide any term by the previous term. Using \(a_2\) and \(a_1\), we get:
- \( r = \frac{a_2}{a_1} = \frac{6}{2} = 3 \)
sequence terms
Sequence terms are the individual elements in the ordered list of a geometric sequence. For the given formula \(a_n = 2 \times (3)^{n-1}\), let's calculate the first four terms to grasp the idea better:
- For \(n = 1\), we have \(a_1 = 2 \times (3)^{1-1} = 2 \times 1 = 2\).
- For \(n = 2\), we get \(a_2 = 2 \times (3)^{2-1} = 2 \times 3 = 6\).
- For \(n = 3\), the term is \(a_3 = 2 \times (3)^{3-1} = 2 \times 9 = 18\).
- Finally, for \(n = 4\), we have \(a_4 = 2 \times (3)^{4-1} = 2 \times 27 = 54\).
exponential functions
Geometric sequences are closely related to exponential functions. In our example, the sequence formula \(a_n = 2 \times (3)^{n-1}\) can be viewed as a discrete form of an exponential function, where the base is 3. Exponential functions have the form \(f(x) = ab^x\), and are essential in understanding phenomena involving growth or decay. In a geometric sequence:
- The base in our formula is 3, indicating exponential growth since the base is greater than 1.
- Each term increases significantly as \ rises due to the multiplication by the base raised to an increasing power.
general formula
The general formula of a geometric sequence provides a rule for finding any term in that sequence without listing all preceding terms. For the sequence \(a_n = 2 \times (3)^{n-1}\), the components are:
- \(2\) - the initial term (also known as the first term when \(n = 1\)).
- \(3\) - the common ratio.
- \(n-1\) - the exponent indicating the term number minus one.
