4.01a Mathematical induction: construct proofs

3 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = 2 ^ { \frac { 1 } { 2 } n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$

3 Let \(S _ { N } = \frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { N } { ( N + 1 ) ! }\). Prove by mathematical induction that, for all positive integers \(N\), $$S _ { N } = 1 - \frac { 1 } { ( N + 1 ) ! }$$

5 It is given that \(y = ( 1 + x ) ^ { 2 } \ln ( 1 + x )\). Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\). Prove by mathematical induction that, for every integer \(n \geqslant 3\), $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { 2 ( n - 3 ) ! } { ( 1 + x ) ^ { n - 2 } }$$

1

1. Use a truth table to prove \(\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm { S } ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm { S } )\).
2. Prove that \(( \mathrm { A } \Rightarrow \mathrm { B } ) \Leftrightarrow ( \sim \mathrm { A } \vee \mathrm { B } )\) and hence use Boolean algebra to prove that $$\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm {~S} ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm {~S} ) .$$
3. A teacher wrote on a report "It is not the case that if Joanna doesn't try then she won't succeed." He meant to say that if Joanna were to try then she would have a chance of success. By letting T be "Joanna will try" and S be "Joanna will succeed", find the real meaning of what the teacher wrote.

1

1. A joke has it that army recruits used to be instructed: "If it moves, salute it. If it doesn't move, paint it." Assume that this instruction has been carried out completely in the local universe, so that everything that doesn't move has been painted.
   1. A recruit encounters something which is not painted. What should he do, and why?
   2. A recruit encounters something which is painted. Do we know what he or she should do? Justify your answer.
2. Use a truth table to prove \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).
3. You are given the following two rules. $$\begin{aligned} & 1 \quad ( a \Rightarrow b ) \Leftrightarrow ( \sim b \Rightarrow \sim a ) \\ & 2 \quad ( x \wedge ( x \Rightarrow y ) ) \Rightarrow y \end{aligned}$$ Use Boolean algebra to prove that \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).

1

1. The Plain English Society presents an annual "Foot in Mouth" award for "a truly baffling comment". In 2004 it was presented to Boris Johnson MP for a comment on the \(12 ^ { \text {th } }\) December 2003 edition of "Have I Got News For You":
   "I could not fail to disagree with you less."
   1. Explain why this can be rewritten as:
      "I could succeed in agreeing with you more."
   2. Rewrite the comment more simply in your own words without changing its meaning.
2. Two switches are to be wired between a mains electricity supply and a light so that when the state of either switch is changed the state of the light changes (i.e. from off to on, or from on to off). Draw a switching circuit to achieve this. The switches are both 2-way switches, thus: \includegraphics[max width=\textwidth, alt={}, center]{88acde67-e22b-478a-9145-48abe931beff-2_127_220_895_1054}
3. Construct a truth table to show the following. $$[ ( a \wedge b ) \vee ( ( \sim a ) \wedge ( \sim b ) ) ] \Leftrightarrow [ ( ( \sim a ) \vee b ) \wedge ( a \vee ( \sim b ) ) ]$$

3 In this question you must show detailed reasoning.
Find the smallest possible positive integers \(m\) and \(n\) such that \(\left( \frac { 64 } { 49 } \right) ^ { - \frac { 3 } { 2 } } = \frac { m } { n }\).

12

1. Show that \(( 1 + i ) ^ { 4 } = - 4\) 12
2. The function f is defined by $$f ( z ) = z ^ { 4 } + 3 z ^ { 2 } - 6 z + 10 \quad z \in \mathbb { C }$$ 12 (b) (i) Show that (1+i) is a root of \(\mathrm { f } ( \mathrm { z } ) = 0\) 12 (b) (ii) Hence write down another root of \(\mathrm { f } ( \mathrm { z } ) = 0\) 12 (b) (iii) One of the linear factors of \(\mathrm { f } ( \mathrm { z } )\) is $$( z - ( 1 + i ) )$$ Write down another linear factor and hence, or otherwise, find a quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
   12 (b) (iv) Find another quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
   12 (b) (v) Hence explain why the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis.

7 Prove by induction that \(2 ^ { n + 1 } + 5 \times 9 ^ { n }\) is divisible by 7 for all integers \(n \geqslant 1\).

4 Prove that \(3 ^ { n } > 10 n\) for all integers \(n \geqslant 4\).

6 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 0 & 1 \end{array} \right)\) where \(a\) is a constant.
Prove by induction that \(\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \text { an } \\ 0 & 1 \end{array} \right)\) for all integers \(n \geqslant 1\).

8 Prove that \(n ! > 2 ^ { n }\) for \(n \geq 4\).

6 The Fibonacci sequence \(\left\{ F _ { n } \right\}\) is defined by \(F _ { 0 } = 0 , F _ { 1 } = 1\) and \(F _ { n } = F _ { n - 1 } + F _ { n - 2 }\) for all \(n \geqslant 2\).

1. Show that \(F _ { n + 5 } = 5 F _ { n + 1 } + 3 F _ { n }\)
2. Prove that \(F _ { n }\) is a multiple of 5 when \(n\) is a multiple of 5 .

3 The sequence \(\left\{ U _ { n } \right\}\) is given by \(U _ { 1 } = 0 , U _ { 2 } = - 1\) and \(U _ { n + 2 } = U _ { n + 1 } + U _ { n } + n - 1\) for \(n \geqslant 1\).

1. List the first seven terms of this sequence. The Fibonacci sequence \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\) is given by \(\mathrm { F } _ { 1 } = 1 , \mathrm {~F} _ { 2 } = 1\) and \(\mathrm { F } _ { \mathrm { n } + 2 } = \mathrm { F } _ { \mathrm { n } + 1 } + \mathrm { F } _ { \mathrm { n } }\) for \(n \geqslant 1\).
2. 1. By comparing the two sequences, give the relationship between \(\mathrm { U } _ { \mathrm { n } }\) and \(\mathrm { F } _ { \mathrm { n } }\).
   2. Show that the relationship found in part (b)(i) holds for all \(n \geqslant 1\).

6 Let \(\mathrm { y } = \mathrm { x } \cosh \mathrm { x }\).
Prove by induction that, for all integers \(n \geqslant 1 , \frac { d ^ { 2 n - 1 } y } { d x ^ { 2 n - 1 } } = x \sinh x + ( 2 n - 1 ) \cosh x\).

8 Prove by induction that \(11 \times 7 ^ { n } - 13 ^ { n } - 1\) is divisible by 3 , for all integers \(n \geqslant 0\).

7 Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9 .

9 Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 5 - 4 r } { 5 ^ { r } } = \frac { n } { 5 ^ { n } }$$

4 In this question you must show detailed reasoning.
Determine the smallest value of \(n\) for which \(\frac { 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } } { 1 + 2 + \ldots + n } > 341\).

9 A function is defined by \(y = f ( t )\) where \(f ( t ) = \ln ( 1 + a t )\) and \(a\) is a constant.

1. By considering \(\frac { d y } { d t } , \frac { d ^ { 2 } y } { d t ^ { 2 } } , \frac { d ^ { 3 } y } { d t ^ { 3 } }\) and \(\frac { d ^ { 4 } y } { d t ^ { 4 } }\), make a conjecture for a general formula for \(\frac { d ^ { n } y } { d t ^ { n } }\) in terms of \(n\) and \(a\) for any integer \(n \geqslant 1\).
2. Use induction to prove the formula conjectured in part (a).
3. In the case where \(\mathrm { f } ( t ) = \ln ( 1 + 2 t )\), find the rate at which the \(6 ^ { \text {th } }\) derivative of \(\mathrm { f } ( t )\) is varying when \(t = \frac { 3 } { 2 }\).

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).

1 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 0 } = 2 , u _ { 1 } = 5\) and \(u _ { n } = \frac { 1 + u _ { n - 1 } } { u _ { n - 2 } }\) for \(n \geqslant 2\).
Prove that the sequence is periodic with period 5.

3 The irrational number \(\phi = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\) plays a significant role in the sequence of Fibonacci numbers given by \(\mathrm { F } _ { 0 } = 0 , \mathrm {~F} _ { 1 } = 1\) and \(\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }\) for \(n \geqslant 1\). Prove by induction that, for each positive integer \(n , \phi ^ { n } = \mathrm { F } _ { \mathrm { n } } \times \phi + \mathrm { F } _ { \mathrm { n } - 1 }\).

8 The sequence \(\left\{ u _ { n } \right\}\) of positive real numbers is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = \frac { 2 u _ { n } + 3 } { u _ { n } + 2 }\) for \(n \geqslant 1\).

1. Prove by induction that \(u _ { n } ^ { 2 } - 3 < 0\) for all positive integers \(n\).
2. By considering \(u _ { n + 1 } - u _ { n }\), use the result of part (a) to show that \(u _ { n + 1 } > u _ { n }\) for all positive integers \(n\). The sequence \(\left\{ u _ { n } \right\}\) has a limit for \(n \rightarrow \infty\).
3. Find the limit of the sequence \(\left\{ u _ { n } \right\}\) as \(n \rightarrow \infty\).
4. Describe as fully as possible the behaviour of the sequence \(\left\{ u _ { n } \right\}\). \section*{END OF QUESTION PAPER}

A rigid body is in equilibrium under the action of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 } \mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act at the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N } \quad \mathbf { r } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { m } \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { N } \quad \mathbf { r } _ { 2 } = ( - \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\)

1. Find the magnitude of \(\mathbf { F } _ { 3 }\)
2. Find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a scalar parameter. \includegraphics[max width=\textwidth, alt={}, center]{cac4dd38-796c-414b-9b80-fe39ab12d41b-11_62_49_2643_1886}