Question #111852
With the help of sequences, calculate √5 correct upto 5 decimals.
1
Expert's answer
2020-04-27T18:32:27-0400

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

Now using the AM-GM inequality, we get;

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

    5(x+(5/x))/2\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 5\sqrt 5 as follows :


Let us consider a recursive sequence given by the formula : xn+1=(xn+5/xn)/2;n2;x1=2x_{n+1}=(x_n+5/x_n)/2 ; \forall n \ge2; x_1=2

    x2=(2+5/2)/2=2.25\implies x_2=(2+5/2)/2=2.25

    x3=(2.25+5/2.25)/2=2.236111\implies x_3=(2.25+5/2.25)/2=2.236111

    x4=(2.236111+5/2.236111)/2=2.236068\implies x_4=(2.236111+5/2.236111)/2=2.236068

    x5=(2.236068+5/2.236068)/2=2.236068\implies x_5=(2.236068+5/2.236068)/2=2.236068


Thus, the sequence converges to the value 2.2360682.236068 .

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

x4=2.2360682.23606x_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.


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