Find equation satisfied by limit

5 The sequence of values given by the iterative formula \(x _ { n + 1 } = \frac { 6 + 8 x _ { n } } { 8 + x _ { n } ^ { 2 } }\) with initial value \(x _ { 1 } = 2\) converges to \(\alpha\).

1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 5 } \left( 4 x _ { n } + \frac { 306 } { x _ { n } ^ { 4 } } \right)$$ with initial value \(x _ { 1 } = 3\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
2. State an equation satisfied by \(\alpha\), and hence show that the exact value of \(\alpha\) is \(\sqrt [ 5 ] { 306 }\).

3 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 2 } { x _ { n } ^ { 3 } }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use this iteration to calculate \(\alpha\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
2. State an equation which is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { 1 } { 2 } \sqrt [ 3 ] { } \left( x _ { n } ^ { 2 } + 6 \right)$$ converges to the value \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places.
2. Find, in the form \(a x ^ { 3 } + b x ^ { 2 } + c = 0\), an equation of which \(\alpha\) is a root.

4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + x _ { n } + 9 } { \left( x _ { n } + 1 \right) ^ { 2 } }$$ with \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Find the value of \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. Determine the exact value of \(\alpha\).

4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 } \left( x _ { n } + \frac { 1 } { x _ { n } ^ { 2 } } \right)$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\).

1. Use this formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.
2. State an equation satisfied by \(\alpha\), and hence find the exact value of \(\alpha\).

8 The equation of a curve is \(y = \ln x + \frac { 2 } { x }\), where \(x > 0\).

1. Find the coordinates of the stationary point of the curve and determine whether it is a maximum or a minimum point.
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 - \ln x _ { n } }$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\). State an equation satisfied by \(\alpha\), and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 3\).
3. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 6 } + 12 x _ { n } } { 3 x _ { n } ^ { 5 } + 8 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use the formula to calculate \(\alpha\) correct to 4 decimal places. Give the result of each iteration to 6 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

8 \includegraphics[max width=\textwidth, alt={}, center]{7fe27759-d014-4bc6-8391-342d9df8280e-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{346e8866-ca23-4ea6-81bf-bf62502a16d1-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } + 4 x _ { n } ^ { - 3 } \right)$$ with initial value \(x _ { 1 } = 1.5\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } } { 3 } + \frac { 4 } { x _ { n } ^ { 2 } }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 7 x _ { n } } { 8 } + \frac { 5 } { 2 x _ { n } ^ { 4 } }$$ with initial value \(x _ { 1 } = 1.7\), converges to \(\alpha\).

1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence show that \(\alpha = \sqrt [ 5 ] { 20 }\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = 2 + \frac { 4 } { x _ { n } ^ { 2 } + 2 x _ { n } + 4 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Determine the value of \(\alpha\) correct to 3 decimal places, giving the result of each iteration to 5 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

1 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 4 } { x _ { n } ^ { 2 } } + \frac { 2 x _ { n } } { 3 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
2. State an equation that is satisfied by \(\alpha\), and hence find the exact value of \(\alpha\).

4 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { x _ { n } } { \ln \left( 2 x _ { n } \right) }$$ converges to the value \(\alpha\).

1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).

2 The sequence defined by $$x _ { 1 } = 3 , \quad x _ { n + 1 } = \sqrt [ 3 ] { 31 - \frac { 5 } { 2 } x _ { n } }$$ converges to the number \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
2. Find an equation of the form \(a x ^ { 3 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers, which has \(\alpha\) as a root.