Originally Posted by

**harish21** For c>0, consider the following quadratic equation:

$\displaystyle x^2$-$\displaystyle x$-$\displaystyle c$=0 , x>0

Define the sequence {$\displaystyle x_n$} recursively fixing $\displaystyle x_1$ > 0 and then, if n is an index for which {$\displaystyle x_n$} has been defined, defining

$\displaystyle x_{n+1}$ = sqrt{c+x_n}

Prove that the sequence {$\displaystyle x_n$} converges monotonically to the solution of the above equation!

--------------------

I tried to attempt this problem by setting $\displaystyle x_1$ = 1, and creating a sequence. It turns out to be a monotonic incereasing sequence. But how do we show that the sequence converges to the solution of the quadratic equation?

Also I am unclear about the starting value of $\displaystyle x_1$. Is it ok to start with $\displaystyle x_1$ = 1, or does the starting value have to be another number?