Problem 29
Question
Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\). $$a_{n}=\frac{1}{1+2 n}$$
Step-by-Step Solution
Verified Answer
The sequence is not arithmetic.
1Step 1: Finding the first term
Substitute \(n = 1\) into the sequence formula \(a_{n} = \frac{1}{1+2n} \) to find \(a_1\). \[a_1 = \frac{1}{1 + 2(1)} = \frac{1}{3}\]
2Step 2: Finding the second term
Substitute \(n = 2\) into the sequence formula \(a_{n} = \frac{1}{1+2n} \) to find \(a_2\). \[a_2 = \frac{1}{1 + 2(2)} = \frac{1}{5}\]
3Step 3: Finding the third term
Substitute \(n = 3\) into the sequence formula \(a_{n} = \frac{1}{1+2n} \) to find \(a_3\). \[a_3 = \frac{1}{1 + 2(3)} = \frac{1}{7}\]
4Step 4: Finding the fourth term
Substitute \(n = 4\) into the sequence formula \(a_{n} = \frac{1}{1+2n} \) to find \(a_4\). \[a_4 = \frac{1}{1 + 2(4)} = \frac{1}{9}\]
5Step 5: Finding the fifth term
Substitute \(n = 5\) into the sequence formula \(a_{n} = \frac{1}{1+2n} \) to find \(a_5\). \[a_5 = \frac{1}{1 + 2(5)} = \frac{1}{11}\]
6Step 6: Checking for arithmetic sequence
To check if the sequence is arithmetic, determine if the difference between consecutive terms is constant. Compute \(a_2 - a_1\), \(a_3 - a_2\), etc. \[a_2 - a_1 = \frac{1}{5} - \frac{1}{3} = \frac{2}{15}\]\[a_3 - a_2 = \frac{1}{7} - \frac{1}{5} = -\frac{2}{35}\]The differences are not equal, so the sequence is not arithmetic.
Key Concepts
Sequence TermsCommon DifferenceStandard Form of a Sequence
Sequence Terms
Understanding the terms of a sequence is crucial when working with series and patterns. In any sequence,
To do so, we substitute successive integer values for \(n\). Starting with \(n = 1\), the first term is found as follows: \[a_1 = \frac{1}{1 + 2(1)} = \frac{1}{3}\]Continuing this process gives us \[a_2 = \frac{1}{5}, \ a_3 = \frac{1}{7}, \ a_4 = \frac{1}{9}, \ a_5 = \frac{1}{11}.\] Each term decreases gradually; this lets us analyze whether the sequence might possess special properties, like being arithmetic or geometric.
- each number has a specific position.
- We label these positions using positive integers, starting from 1.
- The first term is labeled as \(a_1\), the second as \(a_2\), and so forth.
To do so, we substitute successive integer values for \(n\). Starting with \(n = 1\), the first term is found as follows: \[a_1 = \frac{1}{1 + 2(1)} = \frac{1}{3}\]Continuing this process gives us \[a_2 = \frac{1}{5}, \ a_3 = \frac{1}{7}, \ a_4 = \frac{1}{9}, \ a_5 = \frac{1}{11}.\] Each term decreases gradually; this lets us analyze whether the sequence might possess special properties, like being arithmetic or geometric.
Common Difference
The common difference is a key aspect of arithmetic sequences.
It is the fixed amount that each term increases or decreases by to reach the next term.
For a sequence to be arithmetic, this common difference must remain constant between all consecutive terms.To determine whether a sequence is arithmetic, as in the given exercise, you calculate the difference between each successive pair of terms.
Performing calculations:
Thus, no common difference exists here; therefore, we cannot apply the arithmetic sequence formula.
It is the fixed amount that each term increases or decreases by to reach the next term.
For a sequence to be arithmetic, this common difference must remain constant between all consecutive terms.To determine whether a sequence is arithmetic, as in the given exercise, you calculate the difference between each successive pair of terms.
Performing calculations:
- \(a_2 - a_1 = \frac{1}{5} - \frac{1}{3} = -\frac{2}{15}\)
- \(a_3 - a_2 = \frac{1}{7} - \frac{1}{5} = -\frac{2}{35}\)
Thus, no common difference exists here; therefore, we cannot apply the arithmetic sequence formula.
Standard Form of a Sequence
The standard form of an arithmetic sequence is a formula that describes each term based on its position within the sequence.
For arithmetic sequences, the \(n\) th term is represented in the form: \[a_{n} = a + (n-1) d\] Where:
However, in the given exercise, we identified that the sequence \(a_{n} = \frac{1}{1+2n}\) does not have a common difference. Thus, this particular sequence cannot be expressed in the standard form for arithmetic sequences.
This is why it is important to understand the nature of the sequence before attempting to write formulas for other terms.
For arithmetic sequences, the \(n\) th term is represented in the form: \[a_{n} = a + (n-1) d\] Where:
- \(a\) represents the first term of the sequence.
- \(d\) is the common difference.
- \(n\) is the position of the term in the sequence.
However, in the given exercise, we identified that the sequence \(a_{n} = \frac{1}{1+2n}\) does not have a common difference. Thus, this particular sequence cannot be expressed in the standard form for arithmetic sequences.
This is why it is important to understand the nature of the sequence before attempting to write formulas for other terms.
Other exercises in this chapter
Problem 28
Use the Binomial Theorem to expand the expression. $$\left(2 A+B^{2}\right)^{4}$$
View solution Problem 28
Find the \(n\)th term of a sequence whose first several terms are given. \(1,3,5,7, \dots\)
View solution Problem 29
Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$2,6,18,54, \dots$$
View solution Problem 29
Find the indicated terms in the expansion of the given binomial. The first three terms in the expansion of \((x+2 y)^{20}\).
View solution