Edexcel A-Level Mathematics A2 Fp2 2 1 Summation of Simple Finite Series N 1 Questions

Practise Edexcel IAL FP2.2.1 by decomposing rational terms, cancelling adjacent differences and applying prefix formulas to exact finite or infinite sums.

Exam points

rewrite rational terms with partial fractions so adjacent terms cancel
find a sum from r=2 to n and express it in the requested form with constants
use a prefix-sum formula to calculate an exact sum over a restricted range or take a telescoping limit

[Maximum number: 4]

Hence, using the method of differences, show that for all integer values of n,

\sum_{r=1}^{n} \frac{1}{(2 r-1)(2 r+1)(2 r+3)}=\frac{n(n+2)}{a(2 n+b)(2 n+c)}

where a, b and c are integers to be determined.

$\frac18\sum_{r=1}^{n}\left(\frac{1}{2r-1}-\frac{2}{2r+1}+\frac{1}{2r+3}\right)$ $=\frac18\left(1-\frac13-\frac{1}{2n+1}+\frac{1}{2n+3}\right)$ $=\frac18\left(\frac{2(2n+1)(2n+3)-3(2n+3)+3(2n+1)}{3(2n+1)(2n+3)}\right)$ $=\frac{n(n+2)}{3(2n+1)(2n+3)}$

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Edexcel A-Level Mathematics A2 Fp2 2 1 Summation of Simple Finite Series N 1 Questions question 1