Question

question
if $a_1 = 1$ and $a_{n+1} = (a_n)^2 + 3$ then find the value of $a_3$.

Explanation:

Step1: Find \( a_2 \)

Given \( a_1 = 1 \) and the recurrence relation \( a_{n + 1}=(a_n)^2 + 3 \). For \( n = 1 \), substitute \( a_1 = 1 \) into the formula:
\( a_2=(a_1)^2 + 3=(1)^2 + 3 = 1 + 3 = 4 \)

Step2: Find \( a_3 \)

Now, use \( a_2 = 4 \) in the recurrence relation for \( n = 2 \):
\( a_3=(a_2)^2 + 3=(4)^2 + 3 = 16 + 3 = 19 \)

Answer:

\( 19 \)