Problem 8 - Precalculus: Mathematical for Calculus

Question

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

First four terms: \(\frac{1}{2}\), \(-\frac{2}{3}\), \(\frac{3}{4}\), \(-\frac{4}{5}\). 100th term: \(-\frac{100}{101}\).

Step 1: Understanding the Sequence Formula

The sequence is given by the formula \(a_n = (-1)^{n+1} \frac{n}{n+1}\). Here, \((-1)^{n+1}\) will alternate the sign of the terms depending on the value of \(n\). The term \(\frac{n}{n+1}\) will be a fraction where the numerator is \(n\) and the denominator is \(n+1\).

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

Substitute \(n = 1\) into the sequence formula: \(a_1 = (-1)^{1+1} \frac{1}{1+1}\). This simplifies to: \((-1)^2 \cdot \frac{1}{2} = 1 \cdot \frac{1}{2} = \frac{1}{2}\). Thus, \(a_1 = \frac{1}{2}\).

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

Substitute \(n = 2\) into the sequence formula: \(a_2 = (-1)^{2+1} \frac{2}{2+1}\). This simplifies to: \((-1)^3 \cdot \frac{2}{3} = -1 \cdot \frac{2}{3} = -\frac{2}{3}\). Thus, \(a_2 = -\frac{2}{3}\).

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

Substitute \(n = 3\) into the sequence formula: \(a_3 = (-1)^{3+1} \frac{3}{3+1}\). This simplifies to: \((-1)^4 \cdot \frac{3}{4} = 1 \cdot \frac{3}{4} = \frac{3}{4}\). Thus, \(a_3 = \frac{3}{4}\).

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

Substitute \(n = 4\) into the sequence formula: \(a_4 = (-1)^{4+1} \frac{4}{4+1}\). This simplifies to: \((-1)^5 \cdot \frac{4}{5} = -1 \cdot \frac{4}{5} = -\frac{4}{5}\). Thus, \(a_4 = -\frac{4}{5}\).

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

Substitute \(n = 100\) into the sequence formula: \(a_{100} = (-1)^{100+1} \frac{100}{100+1}\). This simplifies to: \((-1)^{101} \cdot \frac{100}{101} = -1 \cdot \frac{100}{101} = -\frac{100}{101}\). Thus, \(a_{100} = -\frac{100}{101}\).

Key Concepts

Alternating Sequences

When you encounter a sequence that switches signs from positive to negative and back again as the terms progress, you're dealing with an alternating sequence. The sequence given in the exercise, with the formula \(a_n = (-1)^{n+1} \frac{n}{n+1}\), is precisely of this nature.

• The expression \((-1)^{n+1}\) is responsible for the alternating signs. It gives us -1 for odd values of \(n\) and 1 for even values.
• This flip-flop of signs makes the sequence "alternate," as each term's sign depends directly on whether \(n\) is even or odd.
• This kind of sequence helps highlight patterns and properties, especially useful in mathematical series and infinite summations where convergence depends on changing signs.

Understanding how this works is key to correctly determining each term’s sign as you move through the sequence.

Sequence Formula

The sequence formula is a mathematical expression that defines the terms of a sequence in order. For the given example, \(a_n = (-1)^{n+1} \frac{n}{n+1}\), the formula tells us how to compute each term based on \(n\), which represents the position of a term within the sequence.

• The component \((-1)^{n+1}\) controls the alternation of the sequence, making every second term switch position from positive to negative and vice versa.
• Meanwhile, the fraction \(\frac{n}{n+1}\) adds a rational nature to each term, beginning just under 1 and decreasing slightly with each successive term. The numerator is \(n\) and the denominator is \(n+1\), ensuring the value of the term is always less than 1.

This formula is designed to be simple yet powerful, enabling you to find any term in the sequence by substituting the appropriate value for \(n\). With these elements, the sequence can reveal rich patterns through the interplay of algebraic arithmetic and alternation.

Nth Term Calculation

Calculating the nth term requires plugging the value of \(n\) into the sequence formula and simplifying it to get the exact value of \(a_n\). This process is what you would follow to find any specific term in the sequence, like the 100th term mentioned in the exercise.
For example, to find the 100th term of the sequence:

• First, identify \(n\) as 100, substitute into the formula: \(a_{100} = (-1)^{100+1} \frac{100}{100+1}\).
• Simplify: Here, \((-1)^{101} = -1\), so the product becomes negative.
• Finally, calculate the fraction: \(\frac{100}{101}\). Putting it all together gives: \(-\frac{100}{101}\).

Calculating the nth term allows dissecting individual elements of the sequence and understanding its behavior pattern. This kind of detailed, term-by-term calculation is essential in verifying the properties and trends within any given mathematical sequence.