A design engineer models the attenuation of a high-frequency signal as it passes through a sequence of 7 capacitive filters. The intensity I I\,I relative to the source is given by I=(1−15x)7\displaystyle I = (1 - \frac{1}{5}x)^7I=(1−51x)7, where x x\,x is a tuning parameter.
Determine the first four terms, in ascending powers of xxx, of the binomial expansion of
(1−15x)7 \left(1 - \frac{1}{5}x\right)^7 (1−51x)7giving each term in its simplest form.
In a modified circuit, the output signal is scaled such that the final intensity is represented by the expansion of
(15x+2)(1−15x)7 (15x + 2)\left(1 - \frac{1}{5}x\right)^7 (15x+2)(1−51x)7Find the coefficient of x3 x^3\,x3 in this new expansion, giving your answer as a fraction in simplest form.
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.
