Problem 11
Question
write the first four terms of each sequence whose general term is given. $$ a_{n}=\frac{(-1)^{n+1}}{2^{n}-1} $$
Step-by-Step Solution
Verified Answer
The first four terms of the sequence are \(a_{1}=1, a_{2}=-\frac{1}{3}, a_{3}=\frac{1}{7}, a_{4}=-\frac{1}{15}\)
1Step 1: Find the First Term/a1
Substitute \(n = 1\) into the formula, we obtain \( a_{1} = \frac{(-1)^{1+1}}{2^{1}-1} = \frac{1}{1} = 1 \)
2Step 2: Find the Second Term/a2
Substitute \(n = 2\) into the formula, which gives \( a_{2} = \frac{(-1)^{2+1}}{2^{2}-1} = \frac{-1}{3} \)
3Step 3: Find the Third Term/a3
Substitute \(n = 3\) into the formula, we get \( a_{3} = \frac{(-1)^{3+1}}{2^{3}-1} = \frac{1}{7} \)
4Step 4: Find the Fourth Term/a4
Substitute \(n = 4\) into the formula, we find \( a_{4} = \frac{(-1)^{4+1}}{2^{4}-1} = \frac{-1}{15} \)
Key Concepts
Arithmetic SequencesGeometric SequencesSequence Terms CalculationAlgebraic Expressions
Arithmetic Sequences
An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a constant to the previous term. This constant is known as the common difference, denoted by 'd'.
For instance, if we have an arithmetic sequence starting with 2, and the common difference is 3, the sequence would progress as follows: 2, 5, 8, 11, and so on. The general term (nth term) of an arithmetic sequence can be found using the formula:
\[ a_n = a_1 + (n - 1)d \]
where \( a_1 \) is the first term and 'n' is the term number. Despite the apparent simplicity, not every sequence with a formula is arithmetic. For instance, the sequence given in the exercise is not an arithmetic sequence because the difference between its terms is not constant.
For instance, if we have an arithmetic sequence starting with 2, and the common difference is 3, the sequence would progress as follows: 2, 5, 8, 11, and so on. The general term (nth term) of an arithmetic sequence can be found using the formula:
\[ a_n = a_1 + (n - 1)d \]
where \( a_1 \) is the first term and 'n' is the term number. Despite the apparent simplicity, not every sequence with a formula is arithmetic. For instance, the sequence given in the exercise is not an arithmetic sequence because the difference between its terms is not constant.
Geometric Sequences
In contrast to arithmetic sequences, a geometric sequence is characterized by each term being multiplied by a constant to get the next term. This constant factor is known as the common ratio, and it's represented by 'r'.
A quick example: starting with 2 and a common ratio of 3, a geometric sequence would go 2, 6, 18, 54, and so forth. To find the nth term of a geometric sequence, the formula used is:
\[ a_n = a_1 \times r^{(n-1)} \]
where \( a_1 \) is the first term. The sequence from the exercise does not fit the geometric pattern either, as the ratio between subsequent terms changes.
A quick example: starting with 2 and a common ratio of 3, a geometric sequence would go 2, 6, 18, 54, and so forth. To find the nth term of a geometric sequence, the formula used is:
\[ a_n = a_1 \times r^{(n-1)} \]
where \( a_1 \) is the first term. The sequence from the exercise does not fit the geometric pattern either, as the ratio between subsequent terms changes.
Sequence Terms Calculation
When it comes to sequence terms calculation, finding specific terms in a sequence involves substituting the term number into the general term formula. As in the provided exercise, we substitute successive values of 'n' to find the corresponding terms.
Here’s a simplified approach to illustrate this: For an nth term formula like \( a_n = 2n + 3 \), calculating the first four terms would be straightforward:
Here’s a simplified approach to illustrate this: For an nth term formula like \( a_n = 2n + 3 \), calculating the first four terms would be straightforward:
- For \( n = 1 \), \( a_1 = 2(1) + 3 = 5 \)
- For \( n = 2 \), \( a_2 = 2(2) + 3 = 7 \)
- For \( n = 3 \), \( a_3 = 2(3) + 3 = 9 \)
- For \( n = 4 \), \( a_4 = 2(4) + 3 = 11 \)
Algebraic Expressions
Finally, understanding algebraic expressions is essential in sequences. An algebraic expression is a combination of numbers, variables, and operations. The sequence term formulas are examples of such expressions, which we manipulate to find specific terms.
For the exercise's sequence \( a_n = \frac{(-1)^{n+1}}{2^n - 1} \), algebra teaches us how exponentiation and subtraction affect the terms. For every 'n', the sign flips due to the exponent of -1, and the increasing power of 2 as 'n' increases creates progressively larger denominators. This level of complexity in the formula emphasizes the need for methodical calculation to determine each term, which makes the sequence neither arithmetic nor geometric but rather a unique type of sequence defined by its own bespoke rule dictated by its algebraic expression.
For the exercise's sequence \( a_n = \frac{(-1)^{n+1}}{2^n - 1} \), algebra teaches us how exponentiation and subtraction affect the terms. For every 'n', the sign flips due to the exponent of -1, and the increasing power of 2 as 'n' increases creates progressively larger denominators. This level of complexity in the formula emphasizes the need for methodical calculation to determine each term, which makes the sequence neither arithmetic nor geometric but rather a unique type of sequence defined by its own bespoke rule dictated by its algebraic expression.
Other exercises in this chapter
Problem 11
In Exercises \(9-16,\) use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given fi
View solution Problem 11
Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (3 x+y)^{3} $$
View solution Problem 11
Use mathematical induction to prove that each statement is true for every positive integer n. \(4+8+12+\dots+4 n=2 n(n+1)\)
View solution Problem 11
Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}-10, a_{1}=30 $$
View solution