Standard summation formulae application

1. Find the value of

$$\sum _ { r = 1 } ^ { 200 } ( r + 1 ) ( r - 1 )$$

5 Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 6 r ^ { 2 } + 2 r \right) = 2 n ^ { 3 } ( n + 1 )$$

3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to find $$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r + 1 ) ,$$ expressing your answer in a fully factorised form.

4 Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + r ^ { 2 } \right) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + 1 )$$

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

5 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } - 3 r ^ { 2 } + r \right)\), giving your answer in a fully factorised form.

1 Use formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$$

4 Find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 )\), giving your answer in a factorised form.

3 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 )\), expressing your answer in a fully factorised form.

4 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } + 1 \right)\), factorising your answer as far as possible.

4 Using the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { n } [ ( r + 1 ) ( r - 2 ) ] = \frac { 1 } { 3 } n \left( n ^ { 2 } - 7 \right)\).

3 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 6 r ^ { 2 } + 2 r \right)\), expressing your answer in a fully factorised form.

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

4 Find \(\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right)\), expressing your answer in a fully factorised form.

4 Find \(\sum _ { r = 1 } ^ { 2 n } \left( 3 r ^ { 2 } - \frac { 1 } { 2 } \right)\), expressing your answer in a fully factorised form.

4 Find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 2 \right)\), expressing your answer in a fully factorised form.

2 Find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 5 \right)\), expressing your answer in a fully factorised form.

1 Find \(\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r - 1 )\), giving your answer in a fully factorised form.

5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r - 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } - 19 \right)\).

4 Using the standard summation formulae, find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 )\). Give your answer in a fully factorised form.

1 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r ( r - 2 )\), factorising your answer as far as possible.

11

1. Evaluate \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).
2. Show that Euler's approximate formula, as given in line 13, gives the exact value of \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).

13 Prove that Euler's approximate formula, as given in line 13, when applied to \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { r } ^ { 2 }\) gives exactly \(\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }\).

4 Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).

1 In this question you must show detailed reasoning.
Determine the value of \(\sum _ { r = 1 } ^ { 50 } r ^ { 2 } ( 16 - r )\).