Problem 9 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The first four terms are 0, 2, 0, 2. The 100th term is 2.

Step 1: Identify the Pattern of the Sequence

Analyze the given sequence formula, \( a_n = 1 + (-1)^n \). Notice that \((-1)^n\) alternates between 1 and -1 depending on whether \(n\) is odd or even. This will affect the term values.

Step 2: Calculate the First Term

Substitute \(n = 1\) into the sequence formula: \(a_1 = 1 + (-1)^1 = 1 - 1 = 0\). Thus, the first term is 0.

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the sequence formula: \(a_2 = 1 + (-1)^2 = 1 + 1 = 2\). Therefore, the second term is 2.

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the sequence formula: \(a_3 = 1 + (-1)^3 = 1 - 1 = 0\). Thus, the third term is 0.

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the sequence formula: \(a_4 = 1 + (-1)^4 = 1 + 1 = 2\). Therefore, the fourth term is 2.

Step 6: Identify the Pattern of Alternation

Notice that the sequence alternates: 0 (odd n), 2 (even n), reflecting the pattern 0, 2, 0, 2, etc.

Step 7: Calculate the 100th Term

Since the pattern is established, we need to determine if the 100th term is affected by an odd or even \(n\). Since 100 is even, \((-1)^{100} = 1\). Therefore, \(a_{100} = 1 + 1 = 2\).

Key Concepts

Alternating Sequences

In mathematics, an alternating sequence is a type of sequence where the terms switch between two different values or patterns as you progress from one term to the next. This alternating pattern is usually observed in terms of sign, magnitude, or both. In the sequence given by the formula \(a_n = 1 + (-1)^n\), the notion of alternation comes from the component \((-1)^n\). This part of the formula switches between 1 and -1 depending on whether the term number \(n\) is odd or even.

When \(n\) is odd, such as 1, 3, 5, etc., \((-1)^n\) equals -1. This creates a term value of \(1 - 1 = 0\).
When \(n\) is even, like 2, 4, 6, etc., \((-1)^n\) equals 1, so the term becomes \(1 + 1 = 2\).
This leads to an alternating cycle in the sequence of 0, 2, 0, 2, and so forth.

Alternating sequences are useful when modeling systems or processes with natural flips or oscillations, such as alternating current in electricity.

Sequence Formula

The sequence formula provides a rule or a relation to generate the terms of a sequence from the position number \(n\). In the given formula \(a_n = 1 + (-1)^n\), each term \(a_n\) is determined based on its position \(n\) in the sequence using this expression:

• 1 represents a baseline value for the sequence.
• \((-1)^n\) introduces alternation that influences the outcome based on \(n\) being odd or even.

The beauty of a sequence formula is its simplicity in generating potentially infinite terms without individual computations for each one. By plugging the position \(n\) into the formula, one can promptly identify the corresponding sequence value. Formulas help identify patterns, making predictions for distant terms more manageable without manual calculations.

Term Calculation

Calculating terms in a sequence involves substituting values into the sequence formula. Let's see this in action:

For the sequence given by \(a_n = 1 + (-1)^n\):

• The first term (\(n = 1\)) is computed as \(a_1 = 1 + (-1)^1 = 0\).
• The second term (\(n = 2\)) results in \(a_2 = 1 + (-1)^2 = 2\).
• Following this, the third term (\(n = 3\)) gives \(a_3 = 1 + (-1)^3 = 0\).
• The fourth term (\(n = 4\)) becomes \(a_4 = 1 + (-1)^4 = 2\).

To find a term far into the sequence, such as the 100th term, apply the formula just the same way:
Since 100 is even, \(a_{100} = 1 + (-1)^{100} = 2\).
Regular practice with term calculations strengthens number manipulation skills and deepens understanding of sequence behaviors. This approach helps you solve not just for any random \(n\), but efficiently find terms far into an infinite series.