Problem 10 - Algebra and Trigonometry

Question

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=(-1)^{n+1} \frac{n}{n+1} $$

Step-by-Step Solution

Answer

The first four terms are \(\frac{1}{2}, -\frac{2}{3}, \frac{3}{4}, -\frac{4}{5}\), and the 100th term is \(-\frac{100}{101}\).

Step 1: Understand the Problem

We need to find the first four terms of the sequence given by the formula \(a_{n} = (-1)^{n+1} \frac{n}{n+1}\) and the 100th term.

Step 2: Find the First Term \(a_1\)

Substitute \(n = 1\) into the formula: \[a_1 = (-1)^{1+1} \frac{1}{1+1} = (-1)^2 \frac{1}{2} = 1 \times \frac{1}{2} = \frac{1}{2}\] So, \(a_1 = \frac{1}{2}\).

Step 3: Find the Second Term \(a_2\)

Substitute \(n = 2\) into the formula: \[a_2 = (-1)^{2+1} \frac{2}{2+1} = (-1)^3 \frac{2}{3} = -1 \times \frac{2}{3} = -\frac{2}{3}\] So, \(a_2 = -\frac{2}{3}\).

Step 4: Find the Third Term \(a_3\)

Substitute \(n = 3\) into the formula: \[a_3 = (-1)^{3+1} \frac{3}{3+1} = (-1)^4 \frac{3}{4} = 1 \times \frac{3}{4} = \frac{3}{4}\] So, \(a_3 = \frac{3}{4}\).

Step 5: Find the Fourth Term \(a_4\)

Substitute \(n = 4\) into the formula: \[a_4 = (-1)^{4+1} \frac{4}{4+1} = (-1)^5 \frac{4}{5} = -1 \times \frac{4}{5} = -\frac{4}{5}\] So, \(a_4 = -\frac{4}{5}\).

Step 6: Find the 100th Term \(a_{100}\)

Substitute \(n = 100\) into the formula: \[a_{100} = (-1)^{100+1} \frac{100}{100+1} = (-1)^{101} \frac{100}{101} = -1 \times \frac{100}{101} = -\frac{100}{101}\] So, \(a_{100} = -\frac{100}{101}\).

Key Concepts

Sequence Formula

A sequence is a list of numbers following a specific pattern. To define this pattern, we use a sequence formula. In our example, the sequence is determined by the formula \(a_n = (-1)^{n+1} \frac{n}{n+1}\). This formula tells us how to calculate each term in the sequence based on its position \(n\).
The sequence formula consists of two parts:

• The sign, \((-1)^{n+1}\), which changes depending on whether \(n\) is odd or even.
• The fraction, \(\frac{n}{n+1}\), which represents each term's value disregarding the sign changes.

By plugging different values of \(n\) into the formula, we can determine the terms of the sequence.
Each position in the sequence corresponds to a unique term dictated by the rules set by the formula, making sequence formulas a powerful tool in mathematics.

Alternating Sequence

An alternating sequence is a type of sequence where the signs of the terms switch back and forth between positive and negative. This alternation is a regular pattern that can depend on whether the position number \(n\) is odd or even.
In our given sequence, \(a_n = (-1)^{n+1} \frac{n}{n+1}\), the alternation is controlled by the expression \((-1)^{n+1}\):

• When \(n+1\) is even, \((-1)^{n+1}\) results in \(1\), leading to a positive term.
• When \(n+1\) is odd, \((-1)^{n+1}\) results in \(-1\), leading to a negative term.

This characteristic of alternating sequences provides a way to create predictable patterns in both mathematics and real-life applications. Understanding this behavior helps in recognizing and working with sequences efficiently.

Term Calculation

Calculating the specific terms of a sequence involves substituting the term number \(n\) into the sequence formula. Following the given formula \(a_n = (-1)^{n+1} \frac{n}{n+1}\), we determine the terms by replacing \(n\) with the desired position number.
To find a term:

• Substitute \(n\) into both \((-1)^{n+1}\) and \(\frac{n}{n+1}\).
• Calculate \((-1)^{n+1}\) to find whether the term will be positive or negative.
• Divide \(n\) by \(n+1\) to get the term's value.

For example, to calculate the 100th term, we input \(n = 100\):
First, \((-1)^{101} = -1\) and then \(\frac{100}{101}\) determines the magnitude. Thus, \(a_{100} = -\frac{100}{101}\).
By consistently applying these steps, you can find any term in the sequence efficiently.