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.
308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.