Question

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

Explanation:

Step1: Find \(a_2\) using \(a_1\)

Given \(a_1 = 2\) and the recurrence relation \(a_{n + 1}=(a_n)^2+3\). For \(n = 1\), we substitute \(a_1\) into the formula to find \(a_2\).
\(a_{2}=(a_1)^2 + 3=(2)^2+3=4 + 3 = 7\)

Step2: Find \(a_3\) using \(a_2\)

Now, we use the value of \(a_2 = 7\) to find \(a_3\). For \(n = 2\) in the recurrence relation \(a_{n + 1}=(a_n)^2+3\), we substitute \(a_2\) into the formula.
\(a_{3}=(a_2)^2+3=(7)^2 + 3=49+3=52\)

Answer:

\(52\)