(i) Find the binomial expansion of (1+3p)−1(1 + 3p)^{-1}(1+3p)−1 up to and including the term in p2p^2p2.
(ii) Show that the first three terms in the binomial expansion of 12−5p\frac{1}{2 - 5p}2−5p1 form a geometric sequence and state the common ratio.
For an automated control system, the sensitivity factor S(p)S(p)S(p) depends on a small perturbation ppp, where S(p)=55p(1+3p)(2−5p)S(p) = \frac{55p}{(1 + 3p)(2 - 5p)}S(p)=(1+3p)(2−5p)55p Express S(p)S(p)S(p) in the form P2−5p+Q1+3p\frac{P}{2 - 5p} + \frac{Q}{1 + 3p}2−5pP+1+3pQ where PPP and QQQ are integers.
(i) Using your answers to parts (a) and (b), find the binomial expansion of G(p)=22p(1+3p)(2−5p)G(p) = \frac{22p}{(1 + 3p)(2 - 5p)}G(p)=(1+3p)(2−5p)22p up to and including the term in p2p^2p2.
(ii) Determine the range of values of ppp for which the binomial expansion of G(p)G(p)G(p) is valid.
Practise Edexcel A Level Maths 4.3 Using Partial Fractions with exam-style questions for A Level Maths. 4 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.