### Home > PC > Chapter 8 > Lesson 8.2.5 > Problem8-142

8-142.

Given the sequence defined by $a_{n+1} = \sqrt { 1 + a _ { n } }$ where $a_{1} = 1$. What is the limit of the sequence $\lim\limits_ { n \rightarrow \infty } a _ { n }$?

1. Using the process as described in problem 8-135, estimate the limit.

$a_{5} =$
$a_{6} =$
$a_{7} =$
$a_{8} =$
$a_{9} =$
$a_{10} =$

2. We can find the limit using algebra by solving the equation $L =\sqrt { 1 + L }$. Solve the equation to find the exact value of the limit.

$L^{2} = 1 + L$
$L^{2} − L − 1 = 0$
Use the quadratic formula!