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).
308 exam-style questions on OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.