The amplitude of a signal in a circuit, S(t)S(t)S(t), can be modeled by the function
S(t)=(2−t4)8 S(t) = \left(2 - \frac{t}{4}\right)^8 S(t)=(2−4t)8where t t\,t represents time in milliseconds.
Find the first 4 terms, in ascending powers of ttt, of the binomial expansion of S(t)S(t)S(t), giving each term in its simplest form.
A noise-cancelling filter F(t)F(t)F(t) is applied such that the resulting signal is given by the product P(t)=F(t)S(t)P(t) = F(t)S(t)P(t)=F(t)S(t). Given that
F(t)=(3+2t)2 F(t) = \left(3 + \frac{2}{t}\right)^2 F(t)=(3+t2)2determine the constant term in the series expansion of P(t)P(t)P(t).
Practise AQA A Level Maths 1.7 D: Sequences and series with exam-style questions for A Level Maths. 267 questions covering 1.7.1 Binomial expansion, 1.7.2 Types of sequence (A-level only), 1.7.3 Sigma notation (A-level only), 1.7.4 Arithmetic sequences and series (A-level only), 1.7.5 Geometric sequences and series (A-level only), and 1.7.6 Sequences and series in modelling (A-level only), matched to the AQA A Level Maths (7357) 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.
