1.01a Proof: structure of mathematical proof and logical steps

5 Prove that, for integer values of \(n\) such that \(0 \leq n < 4\) $$2 ^ { n + 2 } > 3 ^ { n }$$

6 Show that the equation $$\begin{aligned} & \qquad 2 \log _ { 10 } x = \log _ { 10 } 4 + \log _ { 10 } ( x + 8 ) \\ & \text { has exactly one solution. } \\ & \text { Fully justify your answer. } \end{aligned}$$

7

1. Given that \(n\) is a positive integer, express $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ as a single fraction not involving surds.
   7
2. Hence, deduce that $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ is a rational number for all positive integer values of \(n\)

8 Kai is proving that \(n ^ { 3 } - n\) is a multiple of 3 for all positive integer values of \(n\). Kai begins a proof by exhaustion.
Step 1 $$n ^ { 3 } - n = n \left( n ^ { 2 } - 1 \right)$$ Step 2 When \(n = 3 m\), where \(m\) is a \(n ^ { 3 } - n = 3 m \left( 9 m ^ { 2 } - 1 \right)\) non-negative integer which is a multiple of 3 Step 3 When \(n = 3 m + 1\), $$\begin{aligned} & n ^ { 3 } - n = ( 3 m + 1 ) \left( ( 3 m + 1 ) ^ { 2 } - 1 \right) \\ & = ( 3 m + 1 ) \left( 9 m ^ { 2 } \right) \\ & = 3 ( 3 m + 1 ) \left( 3 m ^ { 2 } \right) \end{aligned}$$ Step 5 Therefore \(n ^ { 3 } - n\) is a multiple of 3 for all positive integer values of \(n\) 8

1. Explain the two mistakes that Kai has made after Step 3. Step 4 \section*{which is a multiple of 3
   which is a multiple of 3}
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   别
   all positive integer values of \(n\) \section*{a} \includegraphics[max width=\textwidth, alt={}]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-10_58_49_1037_370} 墐 pount \(\_\_\_\_\) \(\_\_\_\_\) " \(\_\_\_\_\) 8
2. Correct Kai's argument from Step 4 onwards.

6

1. Asif notices that \(24 ^ { 2 } = 576\) and \(2 + 4 = 6\) gives the last digit of 576 He checks two more examples: $$\begin{array} { l c } 27 ^ { 2 } = 729 & 29 ^ { 2 } = 841 \\ 2 + 7 = 9 & 2 + 9 = 11 \\ \text { Last digit } 9 & \text { Last digit } 1 \end{array}$$ Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared. Give a counter example to show that Asif's conclusion is not correct. 6
2. Claire tells Asif that he should look only at the last digit of the number being squared. $$\begin{array} { c c } 27 ^ { 2 } = 729 & 24 ^ { 2 } = 576 \\ 7 ^ { 2 } = 49 & 4 ^ { 2 } = 16 \\ \text { Last digit } 9 & \text { Last digit } 6 \end{array}$$ Using Claire's method determine the last digit of \(23456789 { } ^ { 2 }\) [0pt] [1 mark] 6
3. Given Claire's method is correct, use proof by exhaustion to show that no square number has a last digit of 8

4 In this question you must show detailed reasoning.
Show that \(\frac { 1 } { \sqrt { 10 } + \sqrt { 11 } } + \frac { 1 } { \sqrt { 11 } + \sqrt { 12 } } + \frac { 1 } { \sqrt { 12 } + \sqrt { 13 } } = \frac { 3 } { \sqrt { 10 } + \sqrt { 13 } }\).

7 In this question you must show detailed reasoning.

1. Express \(\ln 3 \times \ln 9 \times \ln 27\) in terms of \(\ln 3\).
2. Hence show that \(\ln 3 \times \ln 9 \times \ln 27 > 6\).

10 Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }\) is an increasing function for all values of \(x\).

3 Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4.

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

1. (i) Prove by counter example that the statement
   "If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number." is false.
   (ii) Use proof by exhaustion to prove that if \(m\) is an integer that is not divisible by 3 , then

$$m ^ { 2 } - 1$$ is divisible by 3

11

1. Prove that $$\sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \frac { 1 } { 2 } \sin ^ { 2 } \theta - \frac { 3 } { 4 } = \frac { 1 } { 4 } \sqrt { 3 } \sin 2 \theta .$$
2. Hence solve the equation $$2 \sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \sin ^ { 2 } \theta = 1 \text { for } - \pi \leqslant \theta \leqslant \pi .$$

5 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where \(n \geqslant 0\).

1. Show that, for all \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
2. Hence prove by induction that, for all positive integers \(n\), $$I _ { n } < n ! .$$

An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]

Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).

1. Find the amount Sean repays in the 21st month. [2]

Over the \(n\) months, he repays a total of £5000.

1. Form an equation in \(n\), and show that your equation may be written as $$n^2 - 150n + 5000 = 0.$$ [3]
2. Solve the equation in part (c). [3]
3. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem. [1]

1. An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n-1)d].$$ [4]

A company made a profit of £54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £\(d\). This model predicts total profits of £619 200 for the 9 years 2001 to 2009 inclusive.

1. Find the value of \(d\). [4]

Using your value of \(d\),

1. find the predicted profit for the year 2011. [2]

The first three terms of an arithmetic series are \(p\), \(5p - 8\), and \(3p + 8\) respectively.

1. Show that \(p = 4\). [2]
2. Find the value of the 40th term of this series. [3]
3. Prove that the sum of the first \(n\) terms of the series is a perfect square. [3]

\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]

Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]

Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]

Simplify \((n + 3)^2 - n^2\). Hence explain why, when \(n\) is an integer, \((n + 3)^2 - n^2\) is never an even number. Given also that \((n + 3)^2 - n^2\) is divisible by \(9\), what can you say about \(n\)? [4]

\(n - 1\), \(n\) and \(n + 1\) are any three consecutive integers.

1. Show that the sum of these integers is always divisible by 3. [1]
2. Find the sum of the squares of these three consecutive integers and explain how this shows that the sum of the squares of any three consecutive integers is never divisible by 3. [3]