5 = x ∗ ( 5 / x ) \sqrt 5=\sqrt{x*(5/x)} 5 = x ∗ ( 5/ x )
Now using the AM-GM inequality, we get;
( x + ( 5 / x ) ) / 2 ≥ x ∗ ( 5 / x ) (x+(5/x))/2 \ge \sqrt{x*(5/x)} ( x + ( 5/ x )) /2 ≥ x ∗ ( 5/ x )
⟹ 5 ≤ ( x + ( 5 / x ) ) / 2 \implies \sqrt 5 \le (x+(5/x))/2 ⟹ 5 ≤ ( 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 5 as follows :
Let us consider a recursive sequence given by the formula : x n + 1 = ( x n + 5 / x n ) / 2 ; ∀ n ≥ 2 ; x 1 = 2 x_{n+1}=(x_n+5/x_n)/2 ; \forall n \ge2; x_1=2 x n + 1 = ( x n + 5/ x n ) /2 ; ∀ n ≥ 2 ; x 1 = 2
⟹ x 2 = ( 2 + 5 / 2 ) / 2 = 2.25 \implies x_2=(2+5/2)/2=2.25 ⟹ x 2 = ( 2 + 5/2 ) /2 = 2.25
⟹ x 3 = ( 2.25 + 5 / 2.25 ) / 2 = 2.236111 \implies x_3=(2.25+5/2.25)/2=2.236111 ⟹ x 3 = ( 2.25 + 5/2.25 ) /2 = 2.236111
⟹ x 4 = ( 2.236111 + 5 / 2.236111 ) / 2 = 2.236068 \implies x_4=(2.236111+5/2.236111)/2=2.236068 ⟹ x 4 = ( 2.236111 + 5/2.236111 ) /2 = 2.236068
⟹ x 5 = ( 2.236068 + 5 / 2.236068 ) / 2 = 2.236068 \implies x_5=(2.236068+5/2.236068)/2=2.236068 ⟹ x 5 = ( 2.236068 + 5/2.236068 ) /2 = 2.236068
Thus, the sequence converges to the value 2.236068 2.236068 2.236068 .
Exact value : 5 = 2.2360679775... \sqrt 5=2.2360679775... 5 = 2.2360679775... ≈ 2.23606 \approx 2.23606 ≈ 2.23606 (correct upto 5 decimal places)
x 4 = 2.236068 ≈ 2.23606 x_4=2.236068 \approx 2.23606 x 4 = 2.236068 ≈ 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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