5=x∗(5/x)
Now using the AM-GM inequality, we get;
(x+(5/x))/2≥x∗(5/x)
⟹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 as follows :
Let us consider a recursive sequence given by the formula : xn+1=(xn+5/xn)/2;∀n≥2;x1=2
⟹x2=(2+5/2)/2=2.25
⟹x3=(2.25+5/2.25)/2=2.236111
⟹x4=(2.236111+5/2.236111)/2=2.236068
⟹x5=(2.236068+5/2.236068)/2=2.236068
Thus, the sequence converges to the value 2.236068 .
Exact value : 5=2.2360679775... ≈2.23606 (correct upto 5 decimal places)
x4=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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