Problem 31 - Algebra and Trigonometry

Question

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 0,2,0,2,0,2, \dots $$

Step-by-Step Solution

Answer

The n-th term is 0 if n is odd, and 2 if n is even.

Step 1: Identify the Sequence Pattern

Observe the given sequence: it alternates between 0 and 2. The pattern repeats every two terms. Thus, the sequence is periodic with period 2.

Step 2: Express the Pattern in Terms of Modulo

The sequence repeats every two terms as 0, 2. Notice that: - If the position is odd (like 1, 3, 5,...), the term is 0. - If the position is even (like 2, 4, 6,...), the term is 2. Express this in terms of modulo: If the term position, n, is odd (n % 2 = 1), the term is 0; if the term position, n, is even (n % 2 = 0), the term is 2.

Step 3: Formulate the n-th Term Expression

Using the observations from Step 2, the n-th term of the sequence can be defined as: \[a_n = \begin{cases} 0, & \text{if } n \equiv 1 \pmod{2} \2, & \text{if } n \equiv 0 \pmod{2}\end{cases}\]

Step 4: Verification

Verify the formula by evaluating it for a few initial terms:- For n = 1 (odd), \(a_1 = 0\).- For n = 2 (even), \(a_2 = 2\).- For n = 3 (odd), \(a_3 = 0\).- For n = 4 (even), \(a_4 = 2\).The formulated expression is consistent with the given sequence.

Key Concepts

Periodic Sequence

A periodic sequence is a sequence that repeats its values in a regular pattern over a certain interval. This interval is known as the period. In the given exercise, the sequence is 0, 2, 0, 2, ..., which repeats every two terms. This makes the period equal to 2. Understanding periodic sequences helps in predicting future values without needing to compute each term individually.

• Periodic sequences are found in many natural phenomena, like wave patterns or seasonal cycles.
• Identifying the period of a sequence makes it easier to write a formula for its terms.
• In simple sequences like the one in this exercise, noticing the repetition aids in describing the sequence efficiently.

Recognizing periodicity involves looking at the sequence and identifying the smallest number of terms after which the sequence starts repeating itself.

Nth Term Formula

The nth term formula is used to determine any term in a sequence without listing all the preceding terms. This is particularly useful in sequences like the one provided, where the pattern is consistent and predictable. In the exercise example, the nth term changes based on whether the position is odd or even, due to the sequence's periodicity.Creating an nth term formula involves:

• Recognizing the pattern or rule that governs the sequence.
• Using mathematical operations, like conditions based on modulo, to express the pattern.
• Setting up a formula that can compute the value of a term at any position n.

For this exercise, the nth term formula is: \[a_n = \begin{cases} 0, & \text{if } n \equiv 1 \pmod{2} \ 2, & \text{if } n \equiv 0 \pmod{2} \end{cases}\] This formula means that for odd positions (like 1, 3, 5), the term is 0, and for even positions (like 2, 4, 6), the term is 2.

Modulo Operation

The modulo operation is a mathematical technique used to find the remainder of division of one number by another. It's a powerful tool in sequences for determining patterns based on position, especially in identifying periodic sequences.In this example, the modulo operation is used to decide if a position is odd or even:

• If \( n \equiv 1 \pmod{2} \), it indicates that n is odd, as the remainder when n is divided by 2 is 1.
• If \( n \equiv 0 \pmod{2} \), it indicates that n is even, with no remainder when n is divided by 2.

Utilizing modulo helps in distinguishing between different term requirements based on position, enabling the formulation of an accurate nth term formula. The modulo operation is widely used in computer science, cryptography, and anywhere repetitive or cyclical patterns are essential to understand.