Use the binomial expansion to expand
(16+2v)−12∣v∣<8 (16 + 2v)^{-\frac{1}{2}} \quad |v| < 8 (16+2v)−21∣v∣<8in ascending powers of vvv, up to and including the term in v2v^2v2, giving each coefficient as a fully simplified fraction.
The pressure PPP in a pneumatic cylinder is modeled by the function
P(v)=12+kv16+2vwhere k is a constant and ∣v∣<8 P(v) = \frac{12 + kv}{\sqrt{16 + 2v}} \quad \text{where } k \text{ is a constant and } |v| < 8 P(v)=16+2v12+kvwhere k is a constant and ∣v∣<8Given that the series expansion of P(v)P(v)P(v), in ascending powers of vvv, is
3+916v+qv2+…where q is a constant 3 + \frac{9}{16}v + qv^2 + \dots \quad \text{where } q \text{ is a constant} 3+169v+qv2+…where q is a constantfind the value of kkk,
find the value 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.