Question

consider the following recursive formula which describes the fibonacci sequence
$a_1 = 1, a_2 = 1$
$a_{n + 1} = a_n + a_{n - 1}$
a. find the first 6 terms of the sequence.
b. is the fibonacci sequence an arithmetic sequence? explain

a. $a_1 = 1, a_2 = 1, a_3 = \square, a_4 = \square, a_5 = \square, a_6 = \square$
(type whole numbers )

Explanation:

Step1: Find \(a_3\)

Using the recursive formula \(a_{n + 1}=a_n + a_{n - 1}\), for \(n = 2\), we have \(a_3=a_2 + a_1\). Substituting \(a_1 = 1\) and \(a_2 = 1\), we get \(a_3=1 + 1 = 2\).

Step2: Find \(a_4\)

For \(n = 3\), the formula gives \(a_4=a_3 + a_2\). Substituting \(a_2 = 1\) and \(a_3 = 2\), we get \(a_4=2 + 1 = 3\).

Step3: Find \(a_5\)

For \(n = 4\), we have \(a_5=a_4 + a_3\). Substituting \(a_3 = 2\) and \(a_4 = 3\), we get \(a_5=3 + 2 = 5\).

Step4: Find \(a_6\)

For \(n = 5\), the formula gives \(a_6=a_5 + a_4\). Substituting \(a_4 = 3\) and \(a_5 = 5\), we get \(a_6=5 + 3 = 8\).

Answer:

\(a_3 = 2\), \(a_4 = 3\), \(a_5 = 5\), \(a_6 = 8\)