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 \(\sum _ { r = 1 } ^ { 100 } ( 2 r + 3 ) ^ { 2 }\).

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.

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.

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.

1 Using any standard results given in the List of Formulae (MF20), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r + 1 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right)$$ for all positive integers \(n\).

Let \(S_N = \sum_{r=1}^{N} (5r + 1)(5r + 6)\) and \(T_N = \sum_{r=1}^{N} \frac{1}{(5r + 1)(5r + 6)}\).

1. Use standard results from the List of Formulae (MF10) to show that $$S_N = \frac{1}{3}N(25N^2 + 90N + 83).$$ [3]
2. Use the method of differences to express \(T_N\) in terms of \(N\). [4]
3. Find \(\lim_{N \to \infty} (N^{-3} S_N T_N)\). [2]

Prove that $$\sum_{r=1}^{n} 6(r^2 - 1) = (n - 1)n(2n + 5).$$ [4]

Prove that \(\sum_{r=1}^{n} (r - 1)(r + 2) = \frac{1}{3} (n - 1)n(n + 4)\). [6]

Use the formulae for \(\sum_{r=1}^{n} r^3\) and \(\sum_{r=1}^{n} r^2\) to find the value of $$\sum_{r=3}^{60} r^2(r - 6)$$ [4 marks]

Use the standard results for \(\sum_{r=1}^n r\) and \(\sum_{r=1}^n r^2\) to show that, for all positive integers \(n\), $$\sum_{r=1}^n (6r^2 + 2r + 1) = n(2n^2 + 4n + 3).$$ [6]

Find \(\sum_{r=1}^{n} (r-1)(r+1)\), giving your answer in a fully factorised form. [6]