Question

Write the first six terms of the sequence. (See Example 1.) \(a_n=-n^2\)

Short answer

The first six terms of the sequence are: -1, -4, -9, -16, -25, -36

Step-by-step solution

Determine the first term

The first term in the sequence is gotten when \(n = 1\), substituting this into the function \(a_n = -n^2\), we get \(a_1 = -(1)^2 = -1\).

Determine the second term

The second term in the sequence is gotten when \(n = 2\), substituting this into the function \(a_n = -n^2\), we get \(a_2 = -(2)^2 = -4\).

Determine the third term

The third term in the sequence is gotten when \(n = 3\), substituting this into the function \(a_n = -n^2\), we get \(a_3 = -(3)^2 = -9\).

Determine the fourth term

The fourth term in the sequence is gotten when \(n = 4\), substituting this into the function \(a_n = -n^2\), we get \(a_4 = -(4)^2 = -16\).

Determine the fifth term

The fifth term in the sequence is gotten when \(n = 5\), substituting this into the function \(a_n = -n^2\), we get \(a_5 = -(5)^2 = -25\).

Determine the sixth term

The sixth term in the sequence is gotten when \(n = 6\), substituting this into the function \(a_n = -n^2\), we get \(a_6 = -(6)^2 = -36\).

Key concepts

Understanding Negative Quadratic Functions

A negative quadratic function takes the form of an equation like \( a_n = -n^2 \). It is a type of quadratic function because it includes \( n^2 \), meaning the variable is squared. When a quadratic function has a negative leading coefficient (in this case, the coefficient is \(-1\)), the graph of the function is a downward-opening parabola. This means that instead of curving upwards, like a happy face, it curves downward, like a sad face.
This implies that as the values for \( n \) (in our sequence) increase, the output values \( a_n \) will decrease. Quadratic functions are significant because they appear very often in real-world situations, from physics problems to financial equations, illustrating how rapidly changing processes behave.
In the context of the sequence \( a_n = -n^2 \), each output is dependent on the square of the input \( n \) but made negative, showing rapid descent in values as \( n \) increases.

Key Features of Sequence Terms

In mathematical sequences, each number is called a term, and they are usually written in order. Sequences can follow various rules or formulas to determine these terms. When you know the rule, you can find any term in the sequence by substituting the appropriate value of \( n \) into your formula.
For example, the sequence described by \( a_n = -n^2 \) produces the terms \( -1, -4, -9, -16, -25, -36 \) as \( n \) goes from 1 to 6. Each of these numbers is a part of the sequence, and they follow a clear pattern dictated by the squaring of \( n \), followed by reversing the sign (due to the negative in the formula).
This method is straightforward and helps predict terms in a sequence without calculating each time. Sequence terms are essential in creating patterns that can be identified and analyzed, providing insight into more significant mathematical and real-world phenomena.

Algebraic Expression Evaluation Skills

Evaluating algebraic expressions is a fundamental skill necessary for solving mathematical problems. This involves substituting numbers for variables and performing the arithmetic operations to find a result. For the sequence \( a_n = -n^2 \), you substitute each value of \( n \) into the expression to find the corresponding term.
For instance, substituting \( n = 1 \) results in \( a_1 = -(1)^2 = -1 \). Similarly, when \( n = 2 \), you calculate \( a_2 = -(2)^2 = -4 \), and this continues for subsequent terms. By systematically evaluating these expressions, you can determine each term of the sequence efficiently.
Mastering this technique is crucial because it helps break down more complex problems into manageable steps, and it's widely applicable across various fields of study in mathematics, including calculus, trigonometry, and beyond. Ensuring accuracy at each step of evaluation is pivotal to solving more comprehensive mathematical challenges.