Suggest and prove formula

8 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = u _ { n } + 2 n + 1\).

1. Show that \(u _ { 4 } = 16\).
2. Hence suggest an expression for \(u _ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 2 }\) and \(\mathbf { A } ^ { 3 }\).
2. Hence suggest a suitable form for the matrix \(\mathbf { A } ^ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 2 & 1 \end{array} \right)\).

1. Show that \(\mathbf { M } ^ { 4 } = \left( \begin{array} { l l } 81 & 0 \\ 80 & 1 \end{array} \right)\).
2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\), where \(n\) is a positive integer.
3. Use induction to prove that your answer to part (ii) is correct.

10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { u _ { n } } { 1 + u _ { n } }\) for \(n \geqslant 1\).

1. Find \(u _ { 2 }\) and \(u _ { 3 }\), and show that \(u _ { 4 } = \frac { 2 } { 7 }\).
2. Hence suggest an expression for \(u _ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = 3 u _ { n } - 2\).

1. Find \(u _ { 2 }\) and \(u _ { 3 }\) and verify that \(\frac { 1 } { 2 } \left( u _ { 4 } - 1 \right) = 27\).
2. Hence suggest an expression for \(u _ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

3 It is given that \(u _ { r } = r \times r !\) for \(r = 1,2,3 , \ldots\). Let \(S _ { n } = u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots + u _ { n }\). Write down the values of $$2 ! - S _ { 1 } , \quad 3 ! - S _ { 2 } , \quad 4 ! - S _ { 3 } , \quad 5 ! - S _ { 4 }$$ Conjecture a formula for \(S _ { n }\). Prove, by mathematical induction, a formula for \(S _ { n }\), for all positive integers \(n\).

9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 2 \end{array} \right)\).

1. By considering \(\mathbf { A } , \mathbf { A } ^ { 2 } , \mathbf { A } ^ { 3 }\) and \(\mathbf { A } ^ { 4 }\) make a conjecture about the form of the matrix \(\mathbf { A } ^ { n }\) in terms of \(n\) for \(n \geqslant 1\).
2. Use induction to prove the conjecture made in part (a).

11

1. It is conjectured that $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = a - \frac { b } { n ! } ,$$ where \(a\) and \(b\) are constants, and \(n\) is an integer such that \(n \geq 2\). By considering particular cases, show that if the conjecture is correct then \(a = b = 1\).
2. Use induction to prove that $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = 1 - \frac { 1 } { n ! } \text { for } n \geq 2 .$$