Problem 2
Question
The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$a_{n}=3(-4)^{n-1}$$
Step-by-Step Solution
Verified Answer
The first five terms are 3, -12, 48, -192, 768; the common ratio is -4.
1Step 1: Calculate the First Term
To find the first term of the sequence, substitute \(n = 1\) into the formula. So, \(a_1 = 3(-4)^{1-1} = 3(-4)^0\). Since \((-4)^0 = 1\), we have \(a_1 = 3\).
2Step 2: Calculate the Second Term
Substitute \(n = 2\) into the formula: \(a_2 = 3(-4)^{2-1} = 3(-4)^1\). Therefore, \(a_2 = 3 \times (-4) = -12\).
3Step 3: Calculate the Third Term
Substitute \(n = 3\) into the formula: \(a_3 = 3(-4)^{3-1} = 3(-4)^2\). Thus, \(a_3 = 3 \times 16 = 48\).
4Step 4: Calculate the Fourth Term
Substitute \(n = 4\) into the formula: \(a_4 = 3(-4)^{4-1} = 3(-4)^3\). Thus, \(a_4 = 3 \times (-64) = -192\).
5Step 5: Calculate the Fifth Term
Substitute \(n = 5\) into the formula: \(a_5 = 3(-4)^{5-1} = 3(-4)^4\). Hence, \(a_5 = 3 \times 256 = 768\).
6Step 6: Find the Common Ratio
The common ratio \(r\) of the sequence is the factor we multiply by to get from one term to the next. As seen in steps 2 and 1, \(r = \frac{a_2}{a_1}\). Hence, \(r = \frac{-12}{3} = -4\).
7Step 7: Plot the Terms on a Graph
To graph the terms, plot the points (1, 3), (2, -12), (3, 48), (4, -192), and (5, 768) on a coordinate plane. Note that this is an exponential graph that oscillates due to alternating positive and negative terms.
Key Concepts
Common RatioExponential GraphSequence Terms
Common Ratio
In the context of a geometric sequence, the common ratio is a crucial concept that helps define the behavior of the sequence. The common ratio, often represented by the symbol \( r \), is the constant factor by which we multiply each term of the sequence to get the next term. In this specific example, where the sequence is defined by the formula \( a_n = 3(-4)^{n-1} \), the common ratio is the number -4.
To find the common ratio, we look at the ratio of two consecutive terms. For the given sequence, going from the first term \( a_1 = 3 \) to the second term \( a_2 = -12 \), we can calculate the common ratio as follows:
\[ r = \frac{a_2}{a_1} = \frac{-12}{3} = -4 \]
This common ratio of -4 tells us that every term is multiplied by -4 to produce the next term. This results in a sequence that alternates between positive and negative values. Understanding the common ratio is essential to predicting future terms and analyzing the sequence's pattern.
To find the common ratio, we look at the ratio of two consecutive terms. For the given sequence, going from the first term \( a_1 = 3 \) to the second term \( a_2 = -12 \), we can calculate the common ratio as follows:
\[ r = \frac{a_2}{a_1} = \frac{-12}{3} = -4 \]
This common ratio of -4 tells us that every term is multiplied by -4 to produce the next term. This results in a sequence that alternates between positive and negative values. Understanding the common ratio is essential to predicting future terms and analyzing the sequence's pattern.
Exponential Graph
An exponential graph is a visual representation of how a geometric sequence's terms relate to each other visually. When plotting the terms of a sequence on a coordinate plane, each term's position can be expressed as a point \((n, a_n)\), with \(n\) being the term number. For the sequence defined by \( a_n = 3(-4)^{n-1} \), the first five plotted terms appear at:
The graph reflects the alternating nature of the sequence, as the values swing from positive to negative and back again. This is a typical feature of sequences with a negative common ratio. As the absolute value of the sequence magnifies rapidly, the exponential graph helps us visualize this dynamic change effectively. Understanding how to graphically represent this sequence is vital for interpreting the underlying mathematical principles.
- (1, 3)
- (2, -12)
- (3, 48)
- (4, -192)
- (5, 768)
The graph reflects the alternating nature of the sequence, as the values swing from positive to negative and back again. This is a typical feature of sequences with a negative common ratio. As the absolute value of the sequence magnifies rapidly, the exponential graph helps us visualize this dynamic change effectively. Understanding how to graphically represent this sequence is vital for interpreting the underlying mathematical principles.
Sequence Terms
Sequence terms are individual elements of a sequence that follow a specific rule or pattern. In a geometric sequence, each term is derived from multiplying the common ratio by the preceding term. The initial term, known as the first term, sets the sequence's starting point.
In the sequence provided by \( a_n = 3(-4)^{n-1} \), the first few terms can be calculated as follows:
In the sequence provided by \( a_n = 3(-4)^{n-1} \), the first few terms can be calculated as follows:
- The first term \(a_1\) is calculated by substituting \(n=1\) into the sequence formula: \( a_1 = 3(-4)^0 = 3 \).
- The second term \(a_2\) follows as \( a_2 = 3(-4)^1 = -12 \).
- Continuing this pattern, the third term \(a_3\) is \( a_3 = 3(-4)^2 = 48 \), and so on.
Other exercises in this chapter
Problem 2
Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$1+4+7+\cdots+(3 n-2)=\frac{n(3 n-1)}{2}$$
View solution Problem 2
Use Pascal's triangle to expand the expression. $$(2 x+1)^{4}$$
View solution Problem 2
Find the first four terms and the 100 th term of the sequence. $$a_{n}=2 n+3$$
View solution Problem 3
A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$a_{n}=\fra
View solution