An iterative formula is shown below. x_(n+1)=(x_(n))/(2)+(1)/(x_(n)) Starting with x_(1)=10 work out the values of x_(3) and x_(4) Give your answers to 1 d.p.

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An iterative formula is shown below. x_(n+1)=(x_(n))/(2)+(1)/(x_(n)) Starting with x_(1)=10 work out the values of x_(3) and x_(4) Give your answers to 1 d.p.

Answer 1

1. Plugging the given value \(10\) for \(x_1\) into the iterative formula gives us the second term, \(x_{2}\): \(x_{2} =\frac{ x_{1} }{2}+\frac{1}{ x_{1} } = \frac{10}{2}+\frac{1}{10} = 5.1\). 2. Now, substitute \(x_{2} = 5.1\) (truncated to 1 decimal place) into the iterative formula to find \(x_{3}\): \(x_{3} =\frac{ x_{2} }{2}+\frac{1}{ x_{2} } = \frac{5}{2}+\frac{1}{5} = 3.0\). 3. Substitute \(x_{3} = 3.0\) (truncated to 1 decimal place) into the iterative equation to find \(x_{4}\): \(x_{4} =\frac{ x_{3} }{2}+\frac{1}{ x_{3} } = \frac{3.0}{2}+\frac{1}{3.0} =2.3\". Therefore, For \( x_{1} =10, x_{3} =3.0 \) and \( x_{4} =2.3 \) each to 1 decimal place.

Answer 2

This question essentially involves a mathematical concept known as iteration, which involves creating new (better) predictions (or terms) according to a mathematical process or formula. The given formula is \[ x_{n+1}=\frac{x_{n}}{2}+\frac{1}{x_{n}}. \] Here, our initial term, \(x_{1} \), is given as 10. The iterative formula indicates using the preceding number to get the next. Starting with \( x_{1} \), you can then use the formula to find \( x_{2} \), then \( x_{3} \), then \( x_{4} \), and so on, with: 1. \( x_{2} =\frac{ x_{1} }{2}+\frac{1}{ x_{1} } \). 2. \( x_{3} =\frac{ x_{2} }{2}+\frac{1}{ x_{2} } \). 3. \( x_{4} =\frac{x_{3}}{2}+\frac{1}{ x_{3} } \). Note Every subsequent value depends on the value of the previous one.

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An iterative formula is shown below. x_(n+1)=(x_(n))/(2)+(1)/(x_(n)) Starting with x_(1)=10 work out the values of x_(3) and x_(4) Give your answers to 1 d.p.

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