Problem 7 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression for the Terms

The sequence is given by the expression \(a_{n} = 1 + (-1)^{n}\). This expression will help us find the specific terms of the sequence based on different values of \(n\).

Step 2: Find the First Term

To find the first term, substitute \(n = 1\) into the expression: \(a_{1} = 1 + (-1)^{1} = 1 - 1 = 0\). So, the first term is 0.

Step 3: Find the Second Term

Substitute \(n = 2\) into the expression: \(a_{2} = 1 + (-1)^{2} = 1 + 1 = 2\). So, the second term is 2.

Step 4: Find the Third Term

Substitute \(n = 3\) into the expression: \(a_{3} = 1 + (-1)^{3} = 1 - 1 = 0\). So, the third term is 0.

Step 5: Find the Fourth Term

Substitute \(n = 4\) into the expression: \(a_{4} = 1 + (-1)^{4} = 1 + 1 = 2\). So, the fourth term is 2.

Step 6: Identify the Pattern

By examining the first four terms \(0, 2, 0, 2\), we observe that the sequence alternates between 0 and 2 for odd and even values of \(n\) respectively.

Step 7: Find the 100th Term

Since the pattern alternates and 100 is an even number, substitute \(n = 100\) into the expression: \(a_{100} = 1 + (-1)^{100} = 1 + 1 = 2\). So, the 100th term is 2.

Key Concepts

sequence

In mathematics, a sequence is a set of numbers arranged in a specific order, following a definite rule. Each number in a sequence is referred to as a 'term'. Sequences can be infinite or finite depending on whether they go on indefinitely or stop at a certain point.
Understanding a sequence involves knowing the mathematical expression that generates the terms. For example, in this exercise, the sequence given is determined by the expression \(a_n = 1 + (-1)^n\). Here, \(n\) represents the position of a term in the sequence, and each term's value is calculated based on its position.
This sequence clearly specifies how each term can be calculated simply by substituting different values of \(n\) into the expression.

• For \(n = 1\), the term is \(0\).
• For \(n = 2\), the term is \(2\).
• For \(n = 3\), the term is \(0\).
• For \(n = 4\), the term is \(2\).

Knowing how sequences work makes it easier to determine any term’s position, even if \(n\) is very large, like the 100th term.

pattern recognition

Pattern recognition in sequences involves identifying regularities or trends in the arrangement of numbers. Recognizing patterns helps to predict future terms of the sequence with ease. In this particular sequence, the pattern is observed by calculating the values of the first few terms.
By looking at the terms 0, 2, 0, 2, you can notice a clear pattern:

• Terms at odd positions (e.g., 1, 3, 5) result in 0.
• Terms at even positions (e.g., 2, 4, 6) yield 2.

Understanding this alternating pattern simplifies the task of finding subsequent terms without repeated substitution. Hence, for problem-solving, once such a pattern is confirmed in a sequence, the pattern can be applied to quickly find unknown terms like the 100th term.

alternating sequences

Alternating sequences demonstrate a specific form of pattern where the values are not the same for consecutive terms. These sequences often switch between two or more fixed numbers or follow a predictable cyclical pattern.
An example of an alternating pattern is evident in this sequence where it alternates between 0 and 2 depending on whether \(n\) is odd or even.

• For odd \(n\), \((-1)^n = -1\), resulting in \(a_n = 0\).
• For even \(n\), \((-1)^n = 1\), resulting in \(a_n = 2\).

In such arrangements, the sequence provides terms that switch periodically based on a set rule. These types of sequences are common in various mathematical contexts and are crucial for their predictive simplicity. Such sequences can be identified early even in visually complex datasets by looking for recurring alternation between specific values.