Answer
Convergent to 2
Work Step by Step
We are given the sequence:
$a_{n+1}=\sqrt{2+a_n}$
$a_0=1$
Compute the values for $a_n$ when n=0,1,2,3,...9:
$a_0=1$
$a_1=\sqrt{2+1}=1.7320508$
$a_2=\sqrt{2+1.7320508}=1.9318517$
$a_3=\sqrt{2+1.9318517}=1.9828897$
$a_4=\sqrt{2+1.9828897}=1.9957178$
$a_5=\sqrt{2+1.9957178}=1.9989292$
$a_6=\sqrt{2+1.9989292}=1.9997323$
$a_7=\sqrt{2+1.9997323}=1.9999331$
$a_8=\sqrt{2+1.9999331}=1.9999833$
$a_9=\sqrt{2+1.9999833}=1.9999958$
We notice that the sequence is convergent and its limit is 2.