Sum from n+1 to 2n or similar range

1 Show that \(\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = \frac { 1 } { 6 } n ( 2 n + 1 ) ( 7 n + 1 )\).

3 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r } = 2 n ^ { 2 } + n$$ Write down the values of \(S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }\). Express \(u _ { r }\) in terms of \(r\), justifying your answer. Find $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } .$$

4 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right)$$

1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = n ^ { 3 }\).
2. Hence show that \(\sum _ { r = n + 1 } ^ { 2 n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = k n ^ { 3 }\) for some integer \(k\).

4

1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 4 r - 3 ) = k n ( n + 1 ) \left( 2 n ^ { 2 } - 1 \right)$$ where \(k\) is a constant.
2. Hence evaluate $$\sum _ { r = 20 } ^ { 40 } r ^ { 2 } ( 4 r - 3 )$$ (2 marks)

8

1. Show that $$\sum _ { r = 1 } ^ { n } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right) = n ( n + p ) ( n + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
2. Hence find the value of $$\sum _ { r = 11 } ^ { 20 } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right)$$ (2 marks)

5 The sum to \(r\) terms, \(S _ { r }\), of a series is given by $$S _ { r } = r ^ { 2 } ( r + 1 ) ( r + 2 )$$ Given that \(u _ { r }\) is the \(r\) th term of the series whose sum is \(S _ { r }\), show that:

1. 1. \(u _ { 1 } = 6\);
   2. \(u _ { 2 } = 42\);
   3. \(\quad u _ { n } = n ( n + 1 ) ( 4 n - 1 )\).
2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } = 3 n ^ { 2 } ( n + 1 ) ( 5 n + 2 )$$

3

1. Use the formulae $$\begin{gathered} \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) \\ \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 } \end{gathered}$$ and $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n \left( n ^ { 2 } - 1 \right) ( 3 n + 2 )$$ (4 marks)
2. Use the result from part (a) to find the value of $$\sum _ { r = 4 } ^ { 11 } r ^ { 2 } ( r - 1 )$$ (3 marks)

3

1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 } r ^ { 2 } ( r + 1 ) ^ { 2 }\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = r ^ { 3 }$$
2. Use the method of differences to show that $$\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )$$

1

1. Given that \(\mathrm { f } ( r ) = ( r - 1 ) r ^ { 2 }\), show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r ( 3 r + 1 )$$
2. Use the method of differences to find the value of $$\sum _ { r = 50 } ^ { 99 } r ( 3 r + 1 )$$ (4 marks)

9

1. Use the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = a n ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are constants to be determined.
   [0pt] [4 marks]
   9
2. Hence, or otherwise, find a fully factorised expression for $$\sum _ { r = n + 1 } ^ { 5 n } r ( r + 3 )$$ $$\mathbf { A } = \left[ \begin{array} { c c } 3 & i - 1 \\ i & 2 \end{array} \right]$$

7 Show that $$\sum _ { r = 11 } ^ { n + 1 } r ^ { 3 } = \frac { 1 } { 4 } \left( n ^ { 2 } + a n + b \right) \left( n ^ { 2 } + a n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be found. \(8 \quad \mathbf { A }\) is a non-singular \(2 \times 2\) matrix and \(\mathbf { A } ^ { \mathrm { T } }\) is the transpose of \(\mathbf { A }\)