Standard +0.3 Part (a) is straightforward algebraic verification requiring common denominators and simplification. Part (b) is a standard telescoping series application where the identity is given—students just need to recognize the pattern and sum the collapsing terms. This is a typical Further Maths method of differences question, slightly easier than average due to the identity being provided rather than derived.
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.
Attempt common denominator of form \(k(n+1)(n+2)\); correct result, no need to show \(a\), \(b\), \(c\) explicitly
## Question 2:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{r+2}{r(r+1)} - \frac{r+3}{(r+1)(r+2)} = \frac{(r+2)^2 - r(r+3)}{r(r+1)(r+2)}$ | M1 | Attempt single fraction with correct denominator |
| $= \frac{r^2+4r+4-r^2-3r}{r(r+1)(r+2)} = \frac{r+4}{r(r+1)(r+2)}$ * | A1* | Correct result, no errors. Must include LHS |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Show telescoping terms from $r=1$ to $r=n$ | M1 | Show sufficient terms to demonstrate cancelling, min 3 at start and 1 at end or 2 at start and 2 at end |
| $\sum_{r=1}^{n} \frac{r+4}{r(r+1)(r+2)} = \frac{3}{2} - \frac{n+3}{(n+1)(n+2)}$ | A1 | Extract the correct 2 remaining terms |
| $= \frac{3(n+1)(n+2)-2n-6}{2(n+1)(n+2)} = \frac{n(3n+7)}{2(n+1)(n+2)}$ | dM1, A1cao | Attempt common denominator of form $k(n+1)(n+2)$; correct result, no need to show $a$, $b$, $c$ explicitly |
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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.
\hfill \mbox{\textit{Edexcel F2 2021 Q2 [6]}}