Method of differences with given identity

A question is this type if and only if it provides or asks to verify an algebraic identity f(r+1) - f(r) = g(r), then uses this to sum Σg(r) by telescoping.

52 questions · Standard +0.4

4.06b Method of differences: telescoping series
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SPS SPS FM 2021 November Q9
7 marks Standard +0.3
  1. Show that $$\frac{1}{9r - 4} - \frac{1}{9r + 5} = \frac{9}{(9r - 4)(9r + 5)}$$ [2 marks]
  2. Hence use the method of differences to find $$\sum_{r=1}^{n} \frac{1}{(9r - 4)(9r + 5)}.$$ [5 marks]
Pre-U Pre-U 9795/1 2011 June Q3
5 marks Challenging +1.2
  1. Express \(\text{f}(r - 1) - \text{f}(r)\) as a single algebraic fraction, where \(\text{f}(r) = \frac{1}{(2r + 1)^2}\). [1]
  2. Hence, using the method of differences, show that $$\sum_{r=1}^{n} \frac{r}{(4r^2 - 1)^2} = \frac{n(n + 1)}{2(2n + 1)^2}$$ for all positive integers \(n\). [4]