(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} 2−5p1form 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)55pExpress 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)22pup 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.
409 exam-style questions on Edexcel A Level Maths Binomial Expansion, covering 4.1 Expanding (1 + x)^n, 4.2 Expanding (a + bx)^n, and 4.3 Using Partial Fractions. Each one has a worked solution and a mark scheme showing where the marks go.