Two linear factors in denominator

1. (a) Express \(\frac { 1 } { ( r + 6 ) ( r + 8 ) }\) in partial fractions.
   (b) Hence show that

$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 6 ) ( r + 8 ) } = \frac { n ( a n + b ) } { 56 ( n + 7 ) ( n + 8 ) }$$ where \(a\) and \(b\) are integers to be found.

1. (a) Express

$$\frac { 1 } { ( n + 3 ) ( n + 5 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all positive integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 5 ) } = \frac { n ( p n + q ) } { 40 ( n + 4 ) ( n + 5 ) }$$ where \(p\) and \(q\) are integers to be determined.
(c) Use the answer to part (b) to determine, as a simplified fraction, the value of $$\frac { 1 } { 9 \times 11 } + \frac { 1 } { 10 \times 12 } + \ldots + \frac { 1 } { 24 \times 26 }$$

11. (a) Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } \equiv \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }\).

1. (a) Express \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) in partial fractions.
   (b) Using your answer to part (a) and the method of differences, show that

$$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { 3 n } { 2 ( 3 n + 2 ) }$$ (c) Evaluate \(\sum _ { r = 100 } ^ { 1000 } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer to 3 significant figures.

3. (a) Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
(b) Hence show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }$$ (c) Evaluate \(\sum _ { r = 10 } ^ { 100 } \frac { 2 } { ( r + 1 ) ( r + 3 ) }\), giving your answer to 3 significant figures.

1. (a) Express \(\frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions.
   (b) Using your answer to (a), find, in terms of \(n\),

$$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$ Give your answer as a single fraction in its simplest form.

1. (a) Express \(\frac { 2 } { ( r + 2 ) ( r + 4 ) }\) in partial fractions.
   (b) Hence show that

$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 2 ) ( r + 4 ) } = \frac { n ( 7 n + 25 ) } { 12 ( n + 3 ) ( n + 4 ) }$$

1. (a) Express \(\frac { 1 } { ( r + 3 ) ( r + 4 ) }\) in partial fractions.
   (b) Hence, using the method of differences, show that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 4 ) } = \frac { n } { a ( n + a ) }$$ where \(a\) is a constant to be found.
(c) Find the exact value of \(\sum _ { r = 15 } ^ { 30 } \frac { 1 } { ( r + 3 ) ( r + 4 ) }\) uestion 1 continued
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2. (a) Express \(\frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 2 n } { 3 ( 2 n + 3 ) }\).

1. (a) Express \(\frac { 1 } { ( r + 6 ) ( r + 8 ) }\) in partial fractions.
   (b) Hence show that

$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 6 ) ( r + 8 ) } = \frac { n ( a n + b ) } { 56 ( n + 7 ) ( n + 8 ) }$$ where \(a\) and \(b\) are integers to be found.

1 Express \(\frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$ Deduce the value of $$\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$

1

1. Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.

1. (a) Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
   (b) Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.
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$$\mathbf { A } = \left( \begin{array} { r r } 4 & - 2
5 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) represents the linear transformation \(M\).
Prove that, for the linear transformation \(M\), there are no invariant lines.
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12. (a) Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } \equiv \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }\).