Standard +0.8 This is a method of differences question requiring partial fractions decomposition, telescoping sum manipulation, and algebraic simplification to match a given form with unknown constants. While the technique is standard for Further Maths, the algebraic manipulation to reach the specific form with three unknowns (a, b, c) requires careful work and is more demanding than routine textbook exercises.
8.
Let
$$S _ { n } = \sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) }$$
where \(n \geq 1\)
Use the method of differences to show that
$$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$
where \(a\), \(b\) and \(c\) are integers. [0pt]
[6 marks]
8.
Let
$$S _ { n } = \sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) }$$
where $n \geq 1$
Use the method of differences to show that
$$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$
where $a$, $b$ and $c$ are integers.\\[0pt]
[6 marks]\\
\hfill \mbox{\textit{SPS SPS FM 2020 Q8 [6]}}