Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.1 Sequences - 12.1 Assess Your Understanding - Page 807: 47

$a_1=\sqrt2$ $a_2=\sqrt{2+\sqrt2}$ $a_3=\sqrt{2+\sqrt{2+\sqrt2}}$ $a_4=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}}$ $a_5=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}}}$

If $a_n=\sqrt{2+a_{n-1}}$ and $a_1=\sqrt2$, then $a_2=\sqrt{2+a_{1}}=\sqrt{2+\sqrt2}$ $a_3=\sqrt{2+a_{2}}=\sqrt{2+\sqrt{2+\sqrt2}}$ $a_4=\sqrt{2+a_{3}}=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}}$ $a_5=\sqrt{2+a_{4}}=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}}}$