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.
