Problem 38 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. \(a_{1}=1 ; \quad a_{n}=n-a_{n-1}\)

Step-by-Step Solution

Answer

The first five terms are 1, 1, 2, 2, 3.

Step 1: Identify the First Term

The first term is given directly in the problem statement. According to the definition, \(a_1 = 1\).

Step 2: Calculate the Second Term

Use the recursive formula to find the second term:\(a_2 = 2 - a_1 = 2 - 1 = 1\). So, \(a_2 = 1\).

Step 3: Calculate the Third Term

Next, use the recursive formula to find the third term:\(a_3 = 3 - a_2 = 3 - 1 = 2\). Therefore, \(a_3 = 2\).

Step 4: Calculate the Fourth Term

Now, calculate the fourth term using the recursive formula:\(a_4 = 4 - a_3 = 4 - 2 = 2\). Thus, \(a_4 = 2\).

Step 5: Calculate the Fifth Term

Finally, use the recursive formula again to find the fifth term:\(a_5 = 5 - a_4 = 5 - 2 = 3\). So, \(a_5 = 3\).

Key Concepts

sequence definition

A sequence is an ordered list of numbers where each number is called a term. In the context of this problem, a sequence is defined recursively. This means that each term is defined based on the previous term(s) in the sequence. The provided recursive definition can help us determine any term in the sequence if we know the previous term(s). In this exercise, the sequence is defined by the formula:

• a first term, denoted as \(a_1\), which is given as 1
• a recursive relation for subsequent terms, represented as \(a_n = n - a_{n-1}\)

recursive formula

A recursive formula allows us to find terms in a sequence using previous terms. In simple terms, it’s a rule that goes from one term to the next. For the given sequence, the recursive formula provided is \(a_n = n - a_{n-1}\). This can be broken down as follows:

• For each term \(a_n\), subtract the previous term (\(a_{n-1}\)) from \(n\)
• Starting with the first term \(a_1 = 1\), apply this rule step-by-step to find subsequent terms

In this way, the sequence is generated by repeatedly applying the recursive relationship to calculate each new term from the previous one.

term calculation

To calculate the terms of a recursively defined sequence, follow the steps precisely. Following is how we can calculate the first five terms of the given sequence:
1. **First Term**: \(a_1\) is directly provided as 1.
2. **Second Term**: Apply the recursive formula: \(a_2 = 2 - a_1 = 2 - 1 = 1\).
3. **Third Term**: Again use the recursive formula: \(a_3 = 3 - a_2 = 3 - 1 = 2\).
4. **Fourth Term**: Using the same process: \(a_4 = 4 - a_3 = 4 - 2 = 2\).
5. **Fifth Term**: Finally, \(a_5 = 5 - a_4 = 5 - 2 = 3\).
By following the recursive formula, each term in the sequence is systematically determined based on the prior terms.

step by step solution

The step-by-step solution breaks down the process of finding the first five terms in the sequence:
1. **Identify the first term**: According to the definition, \(a_1 = 1\).
2. **Calculate the second term**: Use the recursive formula: \(a_2 = 2 - a_1 = 2 - 1 = 1\). So, \(a_2 = 1\).
3. **Calculate the third term**: Apply the recursive formula: \(a_3 = 3 - a_2 = 3 - 1 = 2\). Thus, \(a_3 = 2\).
4. **Calculate the fourth term**: Using the same recursive approach: \(a_4 = 4 - a_3 = 4 - 2 = 2\). Hence, \(a_4 = 2\).
5. **Calculate the fifth term**: Finally, \(a_5 = 5 - a_4 = 5 - 2 = 3\). Meaning, \(a_5 = 3\).
Following these steps ensures that you understand how each term is derived from the recursive formula, providing a clear path from the first term to the fifth term in the sequence.