Standard summation formulae application

2 Find an expression for \(1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + 3 \times 4 ^ { 2 } + \ldots + n ( n + 1 ) ^ { 2 }\) in terms of \(n\). Give your answer in fully factorised form.

4 In this question you must show detailed reasoning.
Determine the value of \(\sum _ { r = 1 } ^ { 100 } ( 2 r + 3 ) ^ { 2 }\).

1 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 )\). Give your answer in a fully factorised form.

6

1. 1. Expand \(( 2 r - 1 ) ^ { 2 }\).
   2. Hence show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
2. Hence find the sum of the squares of the odd numbers between 100 and 200 .

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).

1 Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.

1 Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series \(( 1 \times 2 \times 4 ) + ( 2 \times 3 \times 5 ) + ( 3 \times 4 \times 6 ) + \ldots\),
where \(n\) is a positive integer. Give your answer in fully factorised form.

9 The discrete random variable \(X\) has a uniform distribution over the set of all integers between \(- n\) and \(n\) inclusive, where \(n\) is a positive integer.

1. Given that \(n\) is odd, determine \(\mathrm { P } \left( \mathrm { X } > \frac { 1 } { 2 } \mathrm { n } \right)\), giving your answer as a single fraction in terms of \(n\).
2. Determine the variance of the sum of 10 independent values of \(X\), giving your answer in the form \(\mathrm { an } ^ { 2 } + \mathrm { bn }\), where \(a\) and \(b\) are constants.

7. (a) Find an expression for \(\sum _ { r = 1 } ^ { 2 m } ( r + 2 ) ^ { 2 }\) in the form \(\frac { 1 } { 3 } m \left( a m ^ { 2 } + b m + c \right)\), where \(a , b , c\) are integers whose values are to be determined.
(b) Hence, calculate \(\sum _ { r = 1 } ^ { 20 } ( r + 2 ) ^ { 2 }\).

1. In this question you may assume the results for

$$\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 } \text { and } \sum _ { r = 1 } ^ { n } r$$

1. Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n ^ { 2 } \left( 2 n ^ { 2 } - 1 \right)$$ The sum of the cubes of 10 consecutive positive odd numbers is 99800
2. Use the answer to part (a) to determine the smallest of these 10 consecutive positive odd numbers.

5 \end{array} \right) + \lambda \left( \begin{array} { l } 2
2
3 \end{array} \right)$$

1. Find the acute angle between \(\Pi\) and \(l\).
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
3. \(S\) is the point \(( 4,5 , - 5 )\). Find the shortest distance from \(S\) to \(\Pi\). 2 The complex number \(2 + \mathrm { i }\) is denoted by \(z\).
4. Show that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
5. Plot the following on the Argand diagram in the Printed Answer Booklet.
   • \(z\)
   • \(z ^ { 2 }\)
   • State the relationship between \(\left| z ^ { 2 } \right|\) and \(| z |\).
   • State the relationship between \(\arg \left( z ^ { 2 } \right)\) and \(\arg ( z )\).
   3 In this question you must show detailed reasoning. Use the formula \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to evaluate \(121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }\). 4 You are given that the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x + 4 = 0\) has three roots, \(\alpha , \beta\) and \(\gamma\).
   By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\). 5 In this question you must show detailed reasoning.
   An ant starts from a fixed point \(O\) and walks in a straight line for 1.5 s . Its velocity, \(v \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), can be modelled by \(v = \frac { 1 } { \sqrt { 9 - t ^ { 2 } } }\). By finding the mean value of \(v\) in \(0 \leqslant t \leqslant 1.5\), deduce the average velocity of the ant.

4

1. Find $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 6 r \right)$$ expressing your answer in the form $$k n ( n + 1 ) ( n + p ) ( n + q )$$ where \(k\) is a fraction and \(p\) and \(q\) are integers.
2. It is given that $$S = \sum _ { r = 1 } ^ { 1000 } \left( r ^ { 3 } - 6 r \right)$$ Without calculating the value of \(S\), show that \(S\) is a multiple of 2008 .

8

1. Show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r$$ can be expressed in the form $$k n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(k\) is a rational number and \(a , b\) and \(c\) are integers.
2. Show that there is exactly one positive integer \(n\) for which $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r = 8 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

3 Show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r \right) = k n ( n + 1 ) ( n - 1 )$$ where \(k\) is a rational number.

5 The first four terms of the series \(S\) can be written as $$S = ( 1 \times 2 ) + ( 2 \times 3 ) + ( 3 \times 4 ) + ( 4 \times 5 ) + \ldots$$ 5

1. Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) 5
2. Show that the sum of the first \(n\) terms of \(S\) is equal to $$\frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$$

1 Find an expression for \(1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + 3 \times 4 ^ { 2 } + \ldots + n ( n + 1 ) ^ { 2 }\) in terms of \(n\). Give your answer in fully factorised form.