Sum of Powers Using Standard Formulae

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 9 r ^ { 2 } - 4 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 6 n - 1 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 9 r ^ { 2 } - 4 r + k \left( 2 ^ { r } \right) \right) = 6630$$ (b) find the exact value of the constant \(k\).

1. Show, using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), that

$$\sum _ { r = 1 } ^ { n } 3 ( 2 r - 1 ) ^ { 2 } = n ( 2 n + 1 ) ( 2 n - 1 ) , \text { for all positive integers } n .$$

6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n - 1 ) ( n + a )$$ where \(a\) is an integer to be determined.
(b) Hence find the value of \(n\), where \(n > 1\), that satisfies $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = 10 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

8. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } + 8 r + 3 \right) = \frac { 1 } { 2 } n ( 2 n + 5 ) ( n + 3 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 3 r ^ { 2 } + 8 r + 3 + k \left( 2 ^ { r - 1 } \right) \right) = 3520$$ (b) find the exact value of the constant \(k\).

1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = \frac { 1 } { 3 } n ( n - a ) ( n + a )$$ where \(a\) is a positive integer to be determined.
(b) Hence, or otherwise, state the positive value of \(n\) that satisfies $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = 0$$ Given that $$\sum _ { r = 3 } ^ { 17 } \left( k r ^ { 3 } + r ^ { 2 } - r - 8 \right) = 6710 \quad \text { where } k \text { is a constant }$$ (c) find the exact value of \(k\).

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

4.A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$

1. Write down the first 6 terms of this sequence.
2. Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
3. Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
4. Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
5. Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
6. Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)

6 Using the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) show that $$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$

5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 3 - 4 r ) = \frac { 1 } { 2 } n ( n + 1 ) \left( 1 - 2 n ^ { 2 } \right)\).

1 Use the List of Formulae (MF10) to show that \(\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)\) and \(\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)\) have the same numerical value.

1 Find \(\sum _ { r = 1 } ^ { n } ( 4 r - 3 ) ( 4 r + 1 )\), giving your answer in its simplest form.

7

1. Show that the equation \(4 x ^ { 3 } - x - 540000 = 0\) has a root, \(\alpha\), in the interval \(51 < \alpha < 52\).
2. It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }\).
   1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = \frac { n } { 3 } \left( k n ^ { 2 } - 1 \right)\), where \(k\) is an integer to be found.
   2. Hence show that \(6 S _ { n }\) can be written as the product of three consecutive integers.
3. Find the smallest value of \(N\) for which the sum of the squares of the first \(N\) odd numbers is greater than 180000 .

1 Using standard summation formulae, find \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - 3 r \right)\), giving your answer in fully factorised form.

1 Using standard summation results, show that \(\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } + r \right) \equiv \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 ) ^ { 2 }\).

1 Using standard results given in MF20, show that $$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 2 r ^ { 2 } + 5 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right) ( 3 n + 8 )$$