Method of differences with given identity

2

1. Use standard results from the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( 1 - \mathrm { r } - \mathrm { r } ^ { 2 } \right)\) in terms of \(n\),
   simplifying your answer. simplifying your answer.
2. Show that $$\frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) } = \frac { r + 1 } { ( r + 1 ) ^ { 2 } + 1 } - \frac { r } { r ^ { 2 } + 1 }$$ and hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).

1 Let \(a\) be a positive constant.

1. Use the method of differences to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { ar } + 1 ) ( \mathrm { ar } + \mathrm { a } + 1 ) }\) in terms of \(n\) and \(a\).
2. Find the value of \(a\) for which \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( a r + 1 ) ( a r + a + 1 ) } = \frac { 1 } { 6 }\).

1

1. Use standard results from the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } + 3 r + 1 \right)\) in terms of \(n\), simplifying your answer.
2. Show that $$\frac { 1 } { r ^ { 3 } } - \frac { 1 } { ( r + 1 ) ^ { 3 } } = \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }$$ and hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }\).

1

1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\), show that $$\mathrm { f } ( r - 1 ) - \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) } .$$
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

2. (a) Show that, for \(r > 0\) $$\frac { r + 2 } { r ( r + 1 ) } - \frac { r + 3 } { ( r + 1 ) ( r + 2 ) } = \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) }$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { c ( n + 1 ) ( n + 2 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.

1. (a) Express

$$\frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { n ( n + 2 ) } { a ( 2 n + b ) ( 2 n + c ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.

1. (a) Show that

$$\frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { 1 } { 2 ( r + 1 ) ( r + 2 ) } - \frac { 1 } { 2 ( r + 2 ) ( r + 3 ) }$$ (b) Hence, or otherwise, find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) }$$ giving your answer as a single fraction in its simplest form.

4. (a) Express as a simplified single fraction \(\frac { 1 } { ( r - 1 ) ^ { 2 } } - \frac { 1 } { r ^ { 2 } }\).
(b) Hence prove, by the method of differences, that \(\quad \sum _ { r = 2 } ^ { n } \frac { 2 r - 1 } { r ^ { 2 } ( r - 1 ) ^ { 2 } } = 1 - \frac { 1 } { n ^ { 2 } }\).

1. (a) Express \(\frac { 2 } { 4 r ^ { 2 } - 1 }\) in partial fractions.
   (b) Hence use the method of differences to show that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$

2. (a) Show that, for \(r > 0\) $$r - 3 + \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) }$$ (b) Hence prove, using the method of differences, that $$\sum _ { r = 1 } ^ { n } \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) } = \frac { n \left( n ^ { 2 } + a n + b \right) } { 2 ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.

1. (a) Express as a simplified single fraction \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } }\) (b) Hence prove, by the method of differences, that

$$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$

9

1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\).
2. Hence show that \(\sum _ { r = 1 } ^ { 2 n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { n } { 2 ( 3 n + 1 ) }\).

2

1. Show that \(\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).
2. Hence use the method of differences to find the sum of the series $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$

7

1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) for all integers \(r\).
2. Hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\). Section B (36 marks)

5

1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } \equiv \frac { 5 } { ( 5 r - 2 ) ( 5 r + 3 ) }\) for all integers \(r\).
2. Hence use the method of differences to show that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }\).

5 Use the result \(\frac { 1 } { 5 r - 1 } - \frac { 1 } { 5 r + 4 } \equiv \frac { 5 } { ( 5 r - 1 ) ( 5 r + 4 ) }\) and the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 1 ) ( 5 r + 4 ) }$$ simplifying your answer.

5 Given that \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } \equiv \frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 }\), find \(\sum _ { r = 1 } ^ { 20 } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer as an exact fraction.

5

1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { 30 } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), expressing your answer as a fraction.

5 You are given that \(\frac { 4 } { ( 4 n - 3 ) ( 4 n + 1 ) } \equiv \frac { 1 } { 4 n - 3 } - \frac { 1 } { 4 n + 1 }\). Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 4 r - 3 ) ( 4 r + 1 ) } = \frac { n } { 4 n + 1 }$$

4 Use the identity \(\frac { 1 } { 2 r + 3 } - \frac { 1 } { 2 r + 5 } \equiv \frac { 2 } { ( 2 r + 3 ) ( 2 r + 5 ) }\) and the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 3 ) ( 2 r + 5 ) }\), expressing your answer as a single fraction.

2

1. Verify that $$\frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } } = \frac { 1 } { n \mathrm { e } ^ { n } } - \frac { 1 } { ( n + 1 ) \mathrm { e } ^ { n + 1 } }$$ Let \(S _ { N } = \sum _ { n = 1 } ^ { N } \frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } }\).
2. Express \(S _ { N }\) in terms of \(N\) and e.
   Let \(S = \lim _ { N \rightarrow \infty } S _ { N }\).
3. Find the least value of \(N\) such that \(( N + 1 ) \left( S - S _ { N } \right) < 10 ^ { - 3 }\).

7

1. Show that, for all integers \(r\), $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) }$$ 7
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } = \frac { a n } { b n + c }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
   7
3. Hence, or otherwise, evaluate $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { 99 \times 101 }$$

2

1. Given that $$u _ { r } = \frac { 1 } { 6 } r ( r + 1 ) ( 4 r + 11 )$$ show that $$u _ { r } - u _ { r - 1 } = r ( 2 r + 3 )$$
2. Hence find the sum of the first hundred terms of the series $$1 \times 5 + 2 \times 7 + 3 \times 9 + \ldots + r ( 2 r + 3 ) + \ldots$$

2

1. Given that $$\frac { 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { A } { r ( r + 1 ) } + \frac { B } { ( r + 1 ) ( r + 2 ) }$$ show that \(A = \frac { 1 } { 2 }\) and find the value of \(B\).
2. Use the method of differences to find $$\sum _ { r = 10 } ^ { 98 } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$ giving your answer as a rational number.

3

1. Show that $$\frac { 2 ^ { r + 1 } } { r + 2 } - \frac { 2 ^ { r } } { r + 1 } = \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
2. Hence find $$\sum _ { r = 1 } ^ { 30 } \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$ giving your answer in the form \(2 ^ { n } - 1\), where \(n\) is an integer.