Problem 9 - Precalculus: Mathematics for Calculus

Question

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=\frac{(-1)^{n}}{n^{2}}\)

Step-by-Step Solution

Answer

First four terms: -1, 1/4, -1/9, 1/16; 100th term: 1/10000.

Step 1: Understand the General Formula of the Sequence

The sequence is defined by the general formula \( a_n = \frac{(-1)^n}{n^2} \). This formula gives us the nth term of the sequence.

Step 2: Find the First Term

To find the first term, substitute \( n = 1 \) into the formula: \( a_1 = \frac{(-1)^1}{1^2} = -1 \).

Step 3: Find the Second Term

Substitute \( n = 2 \) into the formula to find the second term: \( a_2 = \frac{(-1)^2}{2^2} = \frac{1}{4} \).

Step 4: Find the Third Term

Substitute \( n = 3 \) into the formula to find the third term: \( a_3 = \frac{(-1)^3}{3^2} = -\frac{1}{9} \).

Step 5: Find the Fourth Term

Substitute \( n = 4 \) into the formula to find the fourth term: \( a_4 = \frac{(-1)^4}{4^2} = \frac{1}{16} \).

Step 6: Find the 100th Term

Substitute \( n = 100 \) into the formula to find the 100th term: \( a_{100} = \frac{(-1)^{100}}{100^2} = \frac{1}{10000} \).

Key Concepts

Nth Term

The idea of the nth term is crucial in understanding sequences. It tells us how to find any term in a sequence without listing all the previous terms.
The nth term is like a "recipe" in mathematics. In our example, the nth term of the sequence is given by the formula \( a_n = \frac{(-1)^n}{n^2} \).

When we know this formula, we can find any term by substituting the value of \( n \) into the formula. This power of prediction is what helps mathematicians analyze and understand patterns easily. For example, to find the 100th term, we substitute 100 in place of \( n \). Following this method, we calculated that the first term is \(-1\), and the 100th term is \(\frac{1}{10000}\). Understanding the nth term streamlines the process of working with long sequences.

General Formula

A general formula in a sequence is the expression that allows you to compute any term. This formula acts as a shortcut, so you don't need to compute every prior term to determine a later one.
For the given sequence, the general formula is \( a_n = \frac{(-1)^n}{n^2} \). This allows us to find any term we need simply by substituting a specific value of \( n \).

A general formula tells us several things about a sequence:

• How each term is calculated
• The behavior of the sequence over time
• The effect of large values of \( n \)

In our example, the formula shows that each term alternates between positive and negative depending on the expression \((-1)^n\). Furthermore, as \( n \) gets larger, each term gets smaller, approaching zero. Knowing the general formula is vital to understanding and predicting sequences.

Alternating Sequence

An alternating sequence is one in which the terms switch signs, going from positive to negative or vice versa. This type of sequence adds an interesting rhythm to mathematical sequences.
In the sequence we are discussing, the formula \( a_n = \frac{(-1)^n}{n^2} \) is responsible for the alternating pattern. The term \((-1)^n\) causes each term's sign to change based on whether \( n \) is odd or even.

Here's how it works:

• If \( n \) is odd, \((-1)^n\) is negative, making the term negative.
• If \( n \) is even, \((-1)^n\) is positive, making the term positive.

This pattern results in a sequence that alternates: for instance, the first four terms are \(-1, \frac{1}{4}, -\frac{1}{9}, \frac{1}{16}\). Recognizing this sign change can help people predict the direction of the series as it progresses.

Mathematical Series

Understanding the concept of a mathematical series involves recognizing the sum of elements from a sequence. When you add up all or part of the terms in a sequence, you have what is known as a series. Mathematical series can be finite or infinite depending on the context.

In the scope of the exercise we're tackling, each term in the sequence gets smaller as \( n \) increases, and thus the series will converge towards a specific value. Often in mathematical contexts, calculations involve determining not only the nth term or formula of a sequence, but also understanding the properties of the sum of those terms.

For sequences like \( a_n = \frac{(-1)^n}{n^2} \), we also consider how the sum of terms behaves. While this exercise doesn't specifically sum the terms, it's an insightful direction for deeper exploration in mathematical series, helping us understand complex concepts like convergence and divergence.