In the calibration of a precision acoustic sensor, the signal response R(s)R(s)R(s) for a sensitivity parameter s s\,s is modeled by the function
R(s)=(2+ks4)5 R(s) = \left( 2 + \frac{ks}{4} \right)^5 R(s)=(2+4ks)5where k k\,k is a non-zero constant.
Find the first 4 terms, in ascending powers of sss, of the binomial expansion of R(s)R(s)R(s). Give each term in its simplest form.
An enhanced signal model is defined by
G(s)=(5+2s)(2+ks4)5 G(s) = (5 + 2s) \left( 2 + \frac{ks}{4} \right)^5 G(s)=(5+2s)(2+4ks)5In the series expansion of G(s)G(s)G(s), the coefficient of s s\,s is equal to the constant term. Find the value of kkk.
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.
