CAIE A-Level Further Math AS 1.3.2 Method of Differences Questions

Practise telescoping sums, partial fractions and the method of differences.

Exam points

apply the method of differences to sums
use partial fractions for telescoping series
solve method of differences problems in context

[Maximum number: 4]

By considering $(r+1)^{2}-r^{2}$ , use the method of differences to prove that

\sum_{r=1}^{n} r=\frac{1}{2} n(n+1) .

$r^{2}+2 r+1-r^{2}=2 r+1$
Expands

$2 \sum_{r=1}^{n} r+n=(n+1)^{2}-1^{2}$
Uses method of differences and sums both sides.

$\Rightarrow 2 \sum_{r=1}^{n} r=n^{2}+n=n(n+1)$
AG.

4

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CAIE A-Level Further Math AS 1.3.2 Method of Differences Questions question 1