Question

the sequence {a_n} is defined by a_1= 1 and a_(n+1) = 1/2( a_n+ (4/a_n) ) for >=1. Assuming that {a_n} converges, find its limit.

Answer 1

$a_{n+1}=\dfrac{a_n+\frac4{a_n}}2$

Assuming convergence of the sequence, we have $a_n\to L$ for some limit $L$ as $n\to\infty$. This also means $a_{n+1}\to L$. So in the above equation, we can substitute and solve for $L$:

$L=\dfrac{L+\frac4L}2$

$\implies2L^2=L^2+4$

$\implies L^2=4\implies L=\pm2$

The limit can only be one of these values; otherwise, the sequence would be divergent. To decide which one is correct, we can observe that the sequence is strictly positive. So $L=2$.