FP2 June 2013 (R) Q3

EdexcelOld spec8 marksSeries

(a) Express \(\dfrac{2}{(r + 1)(r + 3)}\) in partial fractions. (2)

(b) Hence show that \[\sum_{r=1}^{n}\frac{2}{(r + 1)(r + 3)} = \frac{n(5n + 13)}{6(n + 2)(n + 3)}\] (4)

(c) Evaluate \(\displaystyle\sum_{r=10}^{100}\frac{2}{(r + 1)(r + 3)}\), giving your answer to 3 significant figures. (2)

Mark scheme (a)

Scheme	Marks
\(\dfrac{2}{(r + 1)(r + 3)} = \dfrac{A}{r + 1} + \dfrac{B}{r + 3}\) \(2 = A(r + 3) + B(r + 1)\)
\(\dfrac{2}{(r + 1)(r + 3)} = \dfrac{1}{r + 1} - \dfrac{1}{r + 3}\)	M1A1
N.B. for M mark you may see no working. Some will just use the “cover up” method to write the answer directly. This is acceptable.
	(2)

Notes

M1 for attempting the PFs – any valid method

A1 for correct PFs \(\dfrac{2}{(r + 1)(r + 3)} = \dfrac{1}{r + 1} - \dfrac{1}{r + 3}\)

N.B. for M mark you may see no working. Some will just use the “cover up” method to write the answer directly. This is acceptable. Award M1A1 if correct, M0A0 otherwise.

Mark scheme (b)

Scheme	Marks
\(\displaystyle\sum\dfrac{2}{(r + 1)(r + 3)} = \sum\dfrac{1}{r + 1} - \dfrac{1}{r + 3}\)
\(= \left(\dfrac{1}{2} - \dfrac{1}{4}\right) + \left(\dfrac{1}{3} - \dfrac{1}{5}\right) + \left(\dfrac{1}{4} - \dfrac{1}{6}\right) + \ldots\) \(+ \left(\dfrac{1}{n - 1} - \dfrac{1}{n + 1}\right) + \left(\dfrac{1}{n} - \dfrac{1}{n + 2}\right) + \left(\dfrac{1}{n + 1} - \dfrac{1}{n + 3}\right)\)	M1A1ft
\(= \dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{n + 2} - \dfrac{1}{n + 3}\)
\(= \dfrac{5(n + 2)(n + 3) - 6(n + 3) - 6(n + 2)}{6(n + 2)(n + 3)}\)	M1
\(= \dfrac{5n^2 + 25n + 30 - 12n - 30}{6(n + 2)(n + 3)}\)
\(= \dfrac{n(5n + 13)}{6(n + 2)(n + 3)}\) *	A1
	(4)

Notes

If all work in \(r\) instead of \(n\), penalise last A mark only.

M1 for using their PFs to list at least 3 terms at the start and 2 terms at the end so the cancelling can be seen. Must start at \(r = 1\) and end at \(r = n\)

A1ft for correct terms follow through their PFs

M1 for picking out the (4) remaining terms and attempting to form a single fraction (unsimplified numerator with at least 2 terms correct)

A1cso for \(\dfrac{n(5n + 13)}{6(n + 2)(n + 3)}\) * (Check all steps in the working are correct – in particular 3rd line from end in the mark scheme.)

NB: If final answer reached correctly from \(\dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{n + 2} - \dfrac{1}{n + 3}\) (i.e. working shown from this point onwards) give 4/4 (even without individual terms listed)

Mark scheme (c)

Scheme	Marks
\(\displaystyle\sum_{10}^{100} = \sum_{1}^{100} - \sum_{1}^{9}\)	M1
\(= \dfrac{100(500 + 13)}{6 \times 102 \times 103} - \dfrac{9 \times 58}{6 \times 11 \times 12} = \dfrac{1425}{1751} - \dfrac{29}{44} = 0.81382\ldots - 0.65909\ldots\)
\(= 0.1547\ldots = 0.155\)	A1
	(2)
	(8 marks)

Notes

M1 for attempting \(\displaystyle\sum_{1}^{100} - \sum_{1}^{9}\) using the result from (b) (with numbers substituted). Use of \(\displaystyle\sum_{1}^{100} - \sum_{1}^{10}\) scores M0

A1cso for sum = 0.155