Answer to Question #111852 in Real Analysis for Pratibha

Question

With the help of sequences, calculate √5 correct upto 5 decimals.

Answer 1

$\sqrt 5=\sqrt{x*(5/x)}$

Now using the AM-GM inequality, we get;

$(x+(5/x))/2 \ge \sqrt{x*(5/x)}$

$\implies \sqrt 5 \le (x+(5/x))/2$

Thus, using this inequality as an approximation we can correlate the following iterative formula for calculating the value of $\sqrt 5$ as follows :

Let us consider a recursive sequence given by the formula : $x_{n+1}=(x_n+5/x_n)/2 ; \forall n \ge2; x_1=2$

$\implies x_2=(2+5/2)/2=2.25$

$\implies x_3=(2.25+5/2.25)/2=2.236111$

$\implies x_4=(2.236111+5/2.236111)/2=2.236068$

$\implies x_5=(2.236068+5/2.236068)/2=2.236068$

Thus, the sequence converges to the value $2.236068$ .

Exact value : $\sqrt 5=2.2360679775...$ $\approx 2.23606$ (correct upto 5 decimal places)

$x_4=2.236068 \approx 2.23606$ (upto 5 decimal places)

Thus, we obtain the required value upto the required precision in 3 iterations only using the above defined sequence.