Let D(v)D(v)D(v) be a function representing the density of a gas relative to its volume vvv, defined by
D(v)=(2−v8)10 D(v) = \left(2 - \frac{v}{8}\right)^{10} D(v)=(2−8v)10Find the first 4 terms of the binomial expansion of D(v)D(v)D(v) in ascending powers of vvv, giving each coefficient as an integer.
A related physical quantity Q Q\,Q is derived from the density function such that
Q=(3+2v)2D(v) Q = \left(3 + \frac{2}{v}\right)^2 D(v) Q=(3+v2)2D(v)Hence, find the constant term in the series expansion of QQQ.
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.