Use standard formulae to show result

1 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 2 r + 1 \right) = n \left( 2 n ^ { 2 } + 4 n + 3 \right)$$

3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = n ^ { 3 }$$

8

1. Use standard series formulae to show that $$\sum _ { r = 1 } ^ { n } [ r ( r - 1 ) - 1 ] = \frac { 1 } { 3 } n ( n + 2 ) ( n - 2 )$$
2. Prove (*) by mathematical induction.

5

1. Show that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) = \mathrm { n } ^ { 2 }\).
2. Show that \(\frac { \sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) } { \sum _ { \mathrm { r } = \mathrm { n } + 1 } ^ { 2 \mathrm { n } } ( 2 \mathrm { r } - 1 ) } = \mathrm { k }\), where \(k\) is a constant to be determined.

16. (a) Show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 ) = \frac { 1 } { 6 } n ( n + 7 ) ( 2 n + 7 ) .$$ (b) Hence calculate the value of \(\quad \sum _ { r = 10 } ^ { 40 } ( r + 1 ) ( r + 5 )\).
[0pt] [P4 June 2004 Qn 1]

10

1. Prove that $$6 + 3 \sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 2 ) = ( n + 1 ) ( n + 2 ) ( n + 3 )$$ [6 marks]
   10
2. Alex substituted a few values of \(n\) into the expression \(( n + 1 ) ( n + 2 ) ( n + 3 )\) and made the statement:
   "For all positive integers n, $$6 + 3 \sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 2 )$$ is divisible by \(12 . "\) Disprove Alex's statement.
   [0pt] [2 marks]