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

Practise evaluating finite series by decomposing terms, cancelling successive differences, and giving exact formulae or values.

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 formula for S_n to calculate an exact sum over a restricted range

[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