Question

Consider the following recursive formula which describes the Fibonacci sequence.
a 1equals​1, a 2equals1
a Subscript n plus 1equalsa Subscript nplusa Subscript n minus 1
a. Find the first 6 terms of the sequence.
b. Is the Fibonacci sequence an arithmetic​ sequence? Explain.
Question content area bottom
Part 1
a. a 1equals​1, a 2equals​1, a 3equals
  
enter your response here​, a 4equals
  
enter your response here​, a 5equals
  
enter your response here​, a 6equals
  
enter your response here
​(Type whole​ numbers.)

Answer 1

Let's find the first 6 terms of the Fibonacci sequence step by step using the given recursive formula.

1. The first term: \( a_1 = 1 \)
2. The second term: \( a_2 = 1 \)

Now we can use the recursive formula \( a_{n+1} = a_n + a_{n-1} \).

3. The third term:
\[
a_3 = a_2 + a_1 = 1 + 1 = 2
\]

4. The fourth term:
\[
a_4 = a_3 + a_2 = 2 + 1 = 3
\]

5. The fifth term:
\[
a_5 = a_4 + a_3 = 3 + 2 = 5
\]

6. The sixth term:
\[
a_6 = a_5 + a_4 = 5 + 3 = 8
\]

Now we have the first 6 terms:

- \( a_1 = 1 \)
- \( a_2 = 1 \)
- \( a_3 = 2 \)
- \( a_4 = 3 \)
- \( a_5 = 5 \)
- \( a_6 = 8 \)

So the completed answer is:
- \( a_1 = 1 \)
- \( a_2 = 1 \)
- \( a_3 = 2 \)
- \( a_4 = 3 \)
- \( a_5 = 5 \)
- \( a_6 = 8 \)

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**Part b:**

To determine if the Fibonacci sequence is an arithmetic sequence, we can state that:

- An **arithmetic sequence** is defined as a sequence in which the difference between consecutive terms is constant (i.e., \( a_{n+1} - a_n = d \) for a constant \( d \)).

In the Fibonacci sequence, the difference between consecutive terms is not constant. For example:

- \( a_3 - a_2 = 2 - 1 = 1 \)
- \( a_4 - a_3 = 3 - 2 = 1 \)
- \( a_5 - a_4 = 5 - 3 = 2 \)
- \( a_6 - a_5 = 8 - 5 = 3 \)

As we can see, the differences are \( 1, 1, 2, 3 \), which are not constant.

Therefore, the Fibonacci sequence is not an arithmetic sequence.