A researcher models the intensity I I\,I of a signal as it penetrates a specific material of thickness x x\,x cm. The model is given by
I(x)=(1+23x)−3for ∣x∣<32 I(x) = \left(1 + \frac{2}{3}x\right)^{-3} \quad \text{for } |x| < \frac{3}{2} I(x)=(1+32x)−3for ∣x∣<23Find the binomial expansion of I(x)I(x)I(x) in ascending powers of x x\,x up to and including the term in x3x^3x3. Simplify each coefficient fully and present them as exact fractions.
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.