Problem 19 - Algebra and Trigonometry

Question

List the first five terms of each sequence. \(\left\\{c_{n}\right\\}=\left\\{(-1)^{n+1} n^{2}\right\\}\)

Step-by-Step Solution

Answer

1, -4, 9, -16, 25

Step 1: Understand the General Formula

The given sequence is defined by the formula \(c_n = (-1)^{n+1} n^2\). This means each term in the sequence is determined by plugging in the values of \(n\) starting from 1.

Step 2: Calculate the First Term

For \(n = 1\): \[ c_1 = (-1)^{1+1} \times 1^2 = (-1)^2 \times 1 = 1 \]

Step 3: Calculate the Second Term

For \(n = 2\): \[ c_2 = (-1)^{2+1} \times 2^2 = (-1)^3 \times 4 = -4 \]

Step 4: Calculate the Third Term

For \(n = 3\): \[ c_3 = (-1)^{3+1} \times 3^2 = (-1)^4 \times 9 = 9 \]

Step 5: Calculate the Fourth Term

For \(n = 4\): \[ c_4 = (-1)^{4+1} \times 4^2 = (-1)^5 \times 16 = -16 \]

Step 6: Calculate the Fifth Term

For \(n = 5\): \[ c_5 = (-1)^{5+1} \times 5^2 = (-1)^6 \times 25 = 25 \]

Key Concepts

general formula

The general formula given for the sequence is denoted by \( c_n = (-1)^{n+1} n^2 \). This formula helps determine each term in the sequence by substituting different values of \ \ into the equation.
In simpler terms, \( (-1)^{n+1} \) alternates the sign (positive or negative) of each term and \( n^2 \) squares the term number.
This alternation in sign is crucial as it creates a unique pattern within the sequence.
We will use this general formula to find each term step-by-step.

alternating sequence

An alternating sequence is a sequence in which the terms change signs as they progress.
In this sequence, \( (-1)^{n+1} \) is what causes the alternation.
Let's break it down:

• When \( n \) is odd (1, 3, 5,...), \( (-1)^{n+1} \) becomes even, which results in \( (-1)^{even} = 1 \).
• When \( n \) is even (2, 4, 6,...), \( (-1)^{n+1} \) becomes odd, which results in \( (-1)^{odd} = -1 \).

Thus, odd \( n \) gives positive and even \( n \) gives negative results.
This mechanism causes the sequence to flip signs alternatively with each term.

term calculation

To find the sequence terms, we follow these steps:
- Plug the value of \( n \) starting from 1 into the formula.
- Compute the value step-by-step.
Let's calculate the first five terms:

• For \( n = 1 \), \( c_1 = (-1)^{1+1} \times 1^2 = (-1)^2 \times 1 = 1 \).
• For \( n = 2 \), \( c_2 = (-1)^{2+1} \times 2^2 = (-1)^3 \times 4 = -4 \).
• For \( n = 3 \), \( c_3 = (-1)^{3+1} \times 3^2 = (-1)^4 \times 9 = 9 \).
• For \( n = 4 \), \( c_4 = (-1)^{4+1} \times 4^2 = (-1)^5 \times 16 = -16 \).
• For \( n = 5 \), \( c_5 = (-1)^{5+1} \times 5^2 = (-1)^6 \times 25 = 25 \).

Each calculation follows the same basic steps but alternates the sign and squares the number accordingly.

mathematical sequences

A mathematical sequence is a list of numbers that follows a specific pattern.
Sequences can be found everywhere in mathematics and form the basis for many complex theories.
In this exercise, our sequence is both arithmetic (because of the squaring) and geometric (due to the alternating negative and positive signs).
Understanding the pattern in a sequence helps simplify its representation and further calculations.

exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent.
In this sequence, the term number \( n \) is the base, and it is squared, meaning the exponent is 2 (n^2).
Specifically, squaring a number means multiplying it by itself:

• \( 1^2 = 1 \)
• \( 2^2 = 4 \)
• \( 3^2 = 9 \)
• \( 4^2 = 16 \)
• \( 5^2 = 25 \)

This exponentiation creates the magnitude of each term, while the \( (-1)^{n+1} \) factor alternates its sign.
Combining these two operations results in the unique sequence presented.