Questions: Find (a3, a4), and (a5). [ a1=0 a2=1 an=an-1+an-2 ]

Transcript text: Find $a_{3}, a_{4}$, and $a_{5}$. \[ \begin{array}{l} a_{1}=0 \\ a_{2}=1 \\ a_{n}=a_{n-1}+a_{n-2} \end{array} \]

Solution

Solution Steps

To find $a_3$, $a_4$, and $a_5$ in the given sequence, we can use the recurrence relation $a_n = a_{n-1} + a_{n-2}$. Starting with the initial values $a_1 = 0$ and $a_2 = 1$, we can iteratively compute the next terms in the sequence.

Step 1: Initial Values

Given the initial values of the sequence: \[ a_1 = 0 \] \[ a_2 = 1 \]

Step 2: Compute $a_3$

Using the recurrence relation $a_n = a_{n-1} + a_{n-2}$: \[ a_3 = a_2 + a_1 = 1 + 0 = 1 \]

Step 3: Compute $a_4$

Continuing with the recurrence relation: \[ a_4 = a_3 + a_2 = 1 + 1 = 2 \]

Step 4: Compute $a_5$

Finally, using the recurrence relation again: \[ a_5 = a_4 + a_3 = 2 + 1 = 3 \]

Final Answer

\[ \boxed{a_3 = 1} \] \[ \boxed{a_4 = 2} \] \[ \boxed{a_5 = 3} \]

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