Show that, for all positive integers \(r\),
$$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
9
Hence, using the method of differences, show that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) } = \frac { n } { a n + b }$$
where \(a\) and \(b\) are integers to be determined.
9
Hence find the exact value of
$$\sum _ { r = 1001 } ^ { 2000 } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
\(\_\_\_\_\) The curve \(C\) has equation
$$y = \frac { 2 x - 10 } { 3 x - 5 }$$
Figure 1 shows the curve \(C\) with its asymptotes.
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