4.01a Mathematical induction: construct proofs

The position vectors of the points \(P\) and \(Q\) on a rigid body are \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and \(( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\) respectively, relative to a fixed origin \(O\). A force \(\mathbf { F } _ { 1 }\) of magnitude 6 N acts at \(P\) in the direction \(( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\). A force \(\mathbf { F } _ { 2 }\) of magnitude 14 N acts at \(Q\) in the direction \(( 3 \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k } )\). When a force \(\mathbf { F } _ { 3 }\) acts at \(O\), which is also a point on the rigid body, the system of three forces is equivalent to a couple of moment \(\mathbf { G }\)

1. Find \(\mathbf { F } _ { 3 }\)
2. Find G When an additional force \(\mathbf { F } _ { 4 } = ( \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) \mathrm { N }\) also acts at \(O\), the system of four forces is equivalent to a single force \(\mathbf { R }\).
3. Write down \(\mathbf { R }\).
4. Find an equation of the line of action of \(\mathbf { R }\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a parameter.

3 The cubic equation $$2 z ^ { 3 } + p z ^ { 2 } + q z + 16 = 0$$ where \(p\) and \(q\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha = 2 + 2 \sqrt { 3 } \mathrm { i }\).

1. 1. Write down another root, \(\beta\), of the equation.
   2. Find the third root, \(\gamma\).
   3. Find the values of \(p\) and \(q\).
2. 1. Express \(\alpha\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. Show that $$( 2 + 2 \sqrt { 3 } \mathrm { i } ) ^ { n } = 4 ^ { n } \left( \cos \frac { n \pi } { 3 } + \mathrm { i } \sin \frac { n \pi } { 3 } \right)$$
   3. Show that $$\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 ^ { 2 n + 1 } \cos \frac { n \pi } { 3 } + \left( - \frac { 1 } { 2 } \right) ^ { n }$$ where \(n\) is an integer.

7 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { k + 1 } = 2 u _ { k } + 1$$

1. Prove by induction that, for all \(n \geqslant 1\), $$u _ { n } = 3 \times 2 ^ { n - 1 } - 1$$
2. Show that $$\sum _ { r = 1 } ^ { n } u _ { r } = u _ { n + 1 } - ( n + 2 )$$

7

1. Given that $$\mathrm { f } ( k ) = 12 ^ { k } + 2 \times 5 ^ { k - 1 }$$ show that $$\mathrm { f } ( k + 1 ) - 5 \mathrm { f } ( k ) = a \times 12 ^ { k }$$ where \(a\) is an integer.
2. Prove by induction that \(12 ^ { n } + 2 \times 5 ^ { n - 1 }\) is divisible by 7 for all integers \(n \geqslant 1\).

4 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = \frac { 3 } { 4 } \quad u _ { n + 1 } = \frac { 3 } { 4 - u _ { n } }$$ Prove by induction that, for all \(n \geqslant 1\), $$u _ { n } = \frac { 3 ^ { n + 1 } - 3 } { 3 ^ { n + 1 } - 1 }$$

7

1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots + \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$
2. Find the smallest integer \(n\) for which the sum of the series differs from 1 by less than \(10 ^ { - 5 }\).

3 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { n + 1 } = \frac { 5 u _ { n } - 3 } { 3 u _ { n } - 1 }$$ Prove by induction that, for all integers \(n \geqslant 1\), $$u _ { n } = \frac { 3 n + 1 } { 3 n - 1 }$$ (6 marks)

3

1. Express \(( k + 1 ) ^ { 2 } + 5 ( k + 1 ) + 8\) in the form \(k ^ { 2 } + a k + b\), where \(a\) and \(b\) are constants.
2. Prove by induction that, for all integers \(n \geqslant 1\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) \left( \frac { 1 } { 2 } \right) ^ { r - 1 } = 16 - \left( n ^ { 2 } + 5 n + 8 \right) \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$$

5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).

6

1. Using standard summation formulae, show that \(\sum _ { r = 1 } ^ { n } r ( r + 2 ) = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 7 )\).
2. Use induction to prove the result in part (a).

6 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right)\).

1. Calculate \(\mathbf { M } ^ { 2 } , \mathbf { M } ^ { 3 }\) and \(\mathbf { M } ^ { 4 }\).
2. Hence make a conjecture about the matrix \(\mathbf { M } ^ { n }\).
3. Prove your conjecture.

6 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 4 & - 9 \\ 1 & - 2 \end{array} \right)\).

1. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 + 3 n & - 9 n \\ n & 1 - 3 n \end{array} \right)\) for all positive integers \(n\).
2. A student thinks that this formula, when \(n = 0\) and \(n = - 1\), gives the identity matrix and the inverse matrix \(\mathbf { M } ^ { - 1 }\) respectively. Determine whether the student is correct.

5 You are given that \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 2 n + 4\).
Prove by induction that \(u _ { n } = n ^ { 2 } + 3 n + 1\) for all positive integers \(n\).

5 Prove by induction that \(\sum _ { r = 1 } ^ { n } r \times 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\) for all positive integers \(n\).

9 Prove by induction that \(5 ^ { n } + 2 \times 11 ^ { n }\) is divisible by 3 for all positive integers \(n\).

6 Prove by mathematical induction that \(\left( \begin{array} { r l } 2 & 0 \\ - 1 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 1 - 2 ^ { n } & 1 \end{array} \right)\) for all positive integers \(n\). Answer all the questions.
Section B (107 marks)

8 Prove by mathematical induction that \(8 ^ { n } - 3 ^ { n }\) is divisible by 5 for all positive integers \(n\).

8

1. Specify fully the transformation T of the plane associated with the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)\) and \(\lambda\) is a non-zero constant.
2. 1. Find detM.
   2. Deduce two properties of the transformation T from the value of detM.
3. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)\), where \(n\) is a positive integer.
4. Hence specify fully a single transformation which is equivalent to \(n\) applications of the transformation T.

7 Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).

7 Prove that \(\sum _ { r = 1 } ^ { n } \frac { r } { 2 ^ { r - 1 } } = 4 - \frac { n + 2 } { 2 ^ { n - 1 } }\) for all \(n \geqslant 1\).

5 In this question you may assume that if \(p\) and \(q\) are distinct prime numbers and \(\mathbf { p } ^ { \alpha } = \mathbf { q } ^ { \beta }\) where \(\alpha , \beta \in \mathbb { Z }\), then \(\alpha = 0\) and \(\beta = 0\).

1. Prove that it is not possible to find \(a\) and \(b\) for which \(\mathrm { a } , \mathrm { b } \in \mathbb { Z }\) and \(3 = 2 ^ { \frac { \mathrm { a } } { \mathrm { b } } }\).
2. Deduce that \(\log _ { 2 } 3 \notin \mathbb { Q }\).

4. Prove, by mathematical induction, that \(9 ^ { n } + 15\) is a multiple of 8 for all positive integers \(n\).

7. Using mathematical induction, prove that $$\left[ \begin{array} { l l } 2 & 5 \\ 0 & 2 \end{array} \right] ^ { n } = \left[ \begin{array} { c c } 2 ^ { n } & 2 ^ { n - 1 } \times 5 n \\ 0 & 2 ^ { n } \end{array} \right]$$ for all positive integers \(n\).

7. Prove, by mathematical induction, that \(13 ^ { ( 2 n - 1 ) } + 8\) is a multiple of 7 for all positive integers \(n\).
\section*{PLEASE DO NOT WRITE ON THIS PAGE}