Problem 7
Question
For the following exercises, write the first four terms of the sequence. $$ a_{n}=-\frac{16}{n+1} $$
Step-by-Step Solution
Verified Answer
The first four terms are -8, -\(\frac{16}{3}\), -4, and -\(\frac{16}{5}\).
1Step 1: Understand the Problem
We are given a sequence defined by the formula \( a_n = -\frac{16}{n+1} \). Our task is to find the first four terms of this sequence, which correspond to \( n = 1, 2, 3, \) and \( 4 \).
2Step 2: Calculate the First Term
Substitute \( n = 1 \) into the sequence formula to find the first term:\[a_1 = -\frac{16}{1+1} = -\frac{16}{2} = -8\]
3Step 3: Calculate the Second Term
Substitute \( n = 2 \) into the sequence formula to find the second term:\[a_2 = -\frac{16}{2+1} = -\frac{16}{3}\]
4Step 4: Calculate the Third Term
Substitute \( n = 3 \) into the sequence formula to find the third term:\[a_3 = -\frac{16}{3+1} = -\frac{16}{4} = -4\]
5Step 5: Calculate the Fourth Term
Substitute \( n = 4 \) into the sequence formula to find the fourth term:\[a_4 = -\frac{16}{4+1} = -\frac{16}{5}\]
6Step 6: Write the First Four Terms
The first four terms of the sequence are \(-8, -\frac{16}{3}, -4, \) and \(-\frac{16}{5}\).
Key Concepts
Sequence FormulaTerms of a SequenceArithmetic SequenceGeometric Sequence
Sequence Formula
When you come across sequences in mathematics, a sequence formula is like a rulebook. This formula defines how each term of the sequence is calculated, based on its position in the sequence. The sequence formula uses an index, often noted as \(n\), to give order to the elements. Each index value corresponds to a specific term of the sequence.
For example, in the sequence formula given as \(a_{n}=-\frac{16}{n+1}\), \(a_n\) represents the term you are calculating, and \(n+1\) represents the position in the sequence. Understanding the sequence formula allows you to determine any term in the sequence by simply plugging in the value of \(n\).
Sequence formulas are versatile; they can allow you to figure out not just the initial terms but any term within the sequence, hence making them essential for the analysis of sequences.
For example, in the sequence formula given as \(a_{n}=-\frac{16}{n+1}\), \(a_n\) represents the term you are calculating, and \(n+1\) represents the position in the sequence. Understanding the sequence formula allows you to determine any term in the sequence by simply plugging in the value of \(n\).
Sequence formulas are versatile; they can allow you to figure out not just the initial terms but any term within the sequence, hence making them essential for the analysis of sequences.
Terms of a Sequence
Each number in a sequence is called a term. When dealing with sequences, it is essential to recognize that each term maintains a fixed position or index. In our current sequence example, the terms are generated by the formula \(a_n = -\frac{16}{n+1}\). This means for each index \(n\), there is one corresponding term in the sequence.
The first four terms you calculate define how the specific sequence begins; they set a pattern or progression. In our problem, these terms are calculated as follows:
The first four terms you calculate define how the specific sequence begins; they set a pattern or progression. In our problem, these terms are calculated as follows:
- First Term: \(a_1 = -8\)
- Second Term: \(a_2 = -\frac{16}{3}\)
- Third Term: \(a_3 = -4\)
- Fourth Term: \(a_4 = -\frac{16}{5}\)
Arithmetic Sequence
An arithmetic sequence is a kind of sequence where each subsequent term is derived by adding a fixed number, called the common difference, to the previous term. In essence, arithmetic sequences are like evenly spaced steps or increments.
For instance, if we look at a sequence like 2, 4, 6, 8, we find the common difference is 2. Each term is obtained by adding 2 to the preceding term.
However, it's vital to note that not all sequences fit the arithmetic model. In our exercise with the formula \(a_n = -\frac{16}{n+1}\), the terms don't have a fixed addition or subtraction pattern, thus, it is not an arithmetic sequence.
For instance, if we look at a sequence like 2, 4, 6, 8, we find the common difference is 2. Each term is obtained by adding 2 to the preceding term.
However, it's vital to note that not all sequences fit the arithmetic model. In our exercise with the formula \(a_n = -\frac{16}{n+1}\), the terms don't have a fixed addition or subtraction pattern, thus, it is not an arithmetic sequence.
Geometric Sequence
Another common type of sequence is a geometric sequence. Here, each term is found by multiplying the previous term by a constant ratio. Geometric sequences exhibit multiplicative steps and grow or shrink at a steady rate depending on the ratio.
For example, in the sequence 3, 6, 12, 24, we observe a common ratio of 2; each term is twice the previous one.
In contrast, the sequence given in the problem doesn't follow such a multiplicative pattern. Since the form \(a_n = -\frac{16}{n+1}\) depends on division by \(n+1\), it does not have a constant multiplicative ratio. Therefore, it is not classified as a geometric sequence, but rather a more general sequence type.
For example, in the sequence 3, 6, 12, 24, we observe a common ratio of 2; each term is twice the previous one.
In contrast, the sequence given in the problem doesn't follow such a multiplicative pattern. Since the form \(a_n = -\frac{16}{n+1}\) depends on division by \(n+1\), it does not have a constant multiplicative ratio. Therefore, it is not classified as a geometric sequence, but rather a more general sequence type.
Other exercises in this chapter
Problem 7
For the following exercises, find the common ratio for the geometric sequence. $$ -0.125,0.25,-0.5,1,-2, \ldots $$
View solution Problem 7
For the following exercises, find the common difference for the arithmetic sequence provided. $$ \left\\{0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\right\\} $$
View solution Problem 7
Write the first four terms of the sequence. $$a_{n}=-\frac{16}{n+1}$$
View solution Problem 8
Express each description of a sum using summation notation. The sum of \(6 k-5\) from \(k=-2\) to \(k=1\)
View solution