Problem 23
Question
Write the first four terms of the infinite sequence whose nth term is given. \(c_{n}=(-1)^{n}(n-2)^{2}\)
Step-by-Step Solution
Verified Answer
The first four terms are: -1, 0, -1, 4.
1Step 1 - Understand the given formula
The sequence is defined by the formula for the nth term: \[c_{n} = (-1)^{n}(n-2)^{2}\]
2Step 2 - Calculate the first term
Set \(n = 1\) in the formula: \[c_{1} = (-1)^{1}(1-2)^{2}\] Compute: \[ (-1)^{1} = -1 \] and \[ (1-2)^{2} = (-1)^{2} = 1 \] So: \[ c_{1} = -1 \times 1 = -1 \]
3Step 3 - Calculate the second term
Set \(n = 2\) in the formula: \[c_{2} = (-1)^{2}(2-2)^{2}\] Compute: \[ (-1)^{2} = 1 \] and \[ (2-2)^{2} = 0^{2} = 0 \] So: \[ c_{2} = 1 \times 0 = 0 \]
4Step 4 - Calculate the third term
Set \(n = 3\) in the formula: \[c_{3} = (-1)^{3}(3-2)^{2}\] Compute: \[ (-1)^{3} = -1 \] and \[ (3-2)^{2} = 1^{2} = 1 \] So: \[ c_{3} = -1 \times 1 = -1 \]
5Step 5 - Calculate the fourth term
Set \(n = 4\) in the formula: \[c_{4} = (-1)^{4}(4-2)^{2}\] Compute: \[ (-1)^{4} = 1 \] and \[ (4-2)^{2} = 2^{2} = 4 \] So: \[ c_{4} = 1 \times 4 = 4 \]
Key Concepts
Sequence FormulaTerms CalculationExponentiation
Sequence Formula
A sequence formula is a way to represent the nth term of a sequence using a mathematical expression. Each term in the sequence is generated by replacing the variable (usually n) with the position of the term. For the given problem, the sequence formula is \[c_{n} = (-1)^{n}(n-2)^{2}\].
This formula combines elements such as exponentiation and arithmetic operations to determine each term in the sequence.
This formula combines elements such as exponentiation and arithmetic operations to determine each term in the sequence.
Terms Calculation
To find the terms of the sequence, we substitute different values of n into the sequence formula. Let's break it down:
1. For the first term, we set \( n = 1 \). This gives us:
\[c_{1} = (-1)^{1}(1-2)^{2} = -1 \times 1 = -1 \]
2. For the second term, set \( n = 2 \):
\[c_{2} = (-1)^{2}(2-2)^{2} = 1 \times 0 = 0 \]
3. For the third term, set \( n = 3 \):
\[c_{3} = (-1)^{3}(3-2)^{2} = -1 \times 1 = -1 \]
4. For the fourth term, set \( n = 4 \):
\[ c_{4} = (-1)^{4}(4-2)^{2} = 1 \times 4 = 4 \]
This process can be continued to find as many terms as needed.
1. For the first term, we set \( n = 1 \). This gives us:
\[c_{1} = (-1)^{1}(1-2)^{2} = -1 \times 1 = -1 \]
2. For the second term, set \( n = 2 \):
\[c_{2} = (-1)^{2}(2-2)^{2} = 1 \times 0 = 0 \]
3. For the third term, set \( n = 3 \):
\[c_{3} = (-1)^{3}(3-2)^{2} = -1 \times 1 = -1 \]
4. For the fourth term, set \( n = 4 \):
\[ c_{4} = (-1)^{4}(4-2)^{2} = 1 \times 4 = 4 \]
This process can be continued to find as many terms as needed.
Exponentiation
Exponentiation is a mathematical operation that raises a number (the base) to the power of another number (the exponent). In the sequence formula \( c_{n} = (-1)^{n}(n-2)^{2} \), exponentiation appears in two places:
1. \( (-1)^{n} \): This part alternates the sign of each term based on whether n is even or odd. If n is even, \( (-1)^{n} = 1 \). If n is odd, \( (-1)^{n} = -1 \).
2. \( (n-2)^{2} \): This raises \( n-2 \) to the power of 2, squaring the result. Squaring any number (positive or negative) makes it positive. For example, \[ (3-2)^{2} = 1^{2} = 1 \] and \[ (1-2)^{2} = (-1)^{2} = 1 \].
Mastering exponentiation is essential to accurately calculating each term in the sequence.
1. \( (-1)^{n} \): This part alternates the sign of each term based on whether n is even or odd. If n is even, \( (-1)^{n} = 1 \). If n is odd, \( (-1)^{n} = -1 \).
2. \( (n-2)^{2} \): This raises \( n-2 \) to the power of 2, squaring the result. Squaring any number (positive or negative) makes it positive. For example, \[ (3-2)^{2} = 1^{2} = 1 \] and \[ (1-2)^{2} = (-1)^{2} = 1 \].
Mastering exponentiation is essential to accurately calculating each term in the sequence.
Other exercises in this chapter
Problem 22
Write the first four terms of the infinite sequence whose nth term is given. \(a_{n}=\frac{4}{2 n+5}\)
View solution Problem 23
Use the binomial theorem to expand each binomial. $$(x-1)^{7}$$
View solution Problem 24
Use the binomial theorem to expand each binomial. $$(x+1)^{6}$$
View solution Problem 24
Write the first four terms of the infinite sequence whose nth term is given. \(c_{n}=(-1)^{n}(2 n-1)^{2}\)
View solution