Problem 5 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Sequence Formula

The sequence is defined by the formula \(a_n = \frac{(-1)^n}{n^2}\). This means that each term of the sequence is obtained by taking \((-1)^n\) and dividing it by \(n^2\). The \((-1)^n\) part makes the terms alternate in sign, being positive for odd \(n\) and negative for even \(n\).

Step 2: Calculating the First Term

Substitute \(n = 1\) into the formula \(a_n = \frac{(-1)^n}{n^2}\). It becomes \(a_1 = \frac{(-1)^1}{1^2} = \frac{-1}{1} = -1\).

Step 3: Calculating the Second Term

Substitute \(n = 2\) into the formula. It becomes \(a_2 = \frac{(-1)^2}{2^2} = \frac{1}{4}\).

Step 4: Calculating the Third Term

Substitute \(n = 3\) into the formula. It becomes \(a_3 = \frac{(-1)^3}{3^2} = \frac{-1}{9}\).

Step 5: Calculating the Fourth Term

Substitute \(n = 4\) into the formula. It becomes \(a_4 = \frac{(-1)^4}{4^2} = \frac{1}{16}\).

Step 6: Finding the 100th Term

Substitute \(n = 100\) into the formula. It becomes \(a_{100} = \frac{(-1)^{100}}{100^2} = \frac{1}{10000}\) because \((-1)^{100} = 1\) (as 100 is even).

Key Concepts

Alternating Sequence

An alternating sequence is a sequence of numbers in which the terms switch their signs between positive and negative as they progress. This characteristic is crucial because it impacts the overall behavior and convergence of the sequence.
For the given sequence, described by the formula \(a_n = \frac{(-1)^n}{n^2}\), the alternating nature is achieved by the term \((-1)^n\).

• When \(n\) is odd, \((-1)^n = -1\), resulting in a negative term.
• Conversely, when \(n\) is even, \((-1)^n = 1\), making the term positive.

This back-and-forth pattern of positive and negative values is visually and analytically significant in most sequences. It affects how the sums of the sequence behave over large numbers of terms.

Term Calculation

Calculating the terms in a sequence involves substituting the term index \(n\) into the sequence's formula. For example, let's break down how to find terms in our sequence formula, \(a_n = \frac{(-1)^n}{n^2}\).

• Substitute the desired term number into the place of \(n\) in the formula.
• Solve the formula step-by-step: first calculate \((-1)^n\) to determine the sign, then evaluate \(n^2\) and perform the division.

Here's a rundown of our calculations using this method:

• For the 1st term, \(n=1\): \(a_1 = \frac{(-1)^1}{1^2} = -1\)
• For the 2nd term, \(n=2\): \(a_2 = \frac{(-1)^2}{2^2} = \frac{1}{4}\)
• Continue this exact process for any \(n\) to find consecutive terms efficiently.

Following this simple calculation guide ensures you accurately compute any term in a sequence.

Nth Term

Finding the nth term of a sequence is a fundamental task when analyzing sequences. It involves a basic understanding of the sequence's rule, in this case \(a_n = \frac{(-1)^n}{n^2}\).
Knowing how to extract the nth term is particularly useful for evaluating both specific and general behaviors within a sequence.

• To find the nth term, identify the value of \(n\) and substitute it into the formula.
• Compute \((-1)^n\) to decide the term's sign.
• Calculate \(n^2\), and then divide to finalize the term's value.

Understanding the process to find the nth term enables one to examine distant elements in a sequence efficiently. For example:

• To find the 100th term, substitute \(n = 100\): \(a_{100} = \frac{(-1)^{100}}{100^2} = \frac{1}{10000}\).

This result shows that at large values of \(n\), terms tend to zero, spotlighting the diminishing magnitude over increasing indices.