A precise micro-sensor measures the relative volume V V\,V of a specialized crystal alloy as it undergoes temperature changes TTT. The relationship is modelled by the formula V(T)=1+pT3V(T) = \sqrt[3]{1 + pT}V(T)=31+pT, where p p\,p is a constant. The binomial expansion of V(T)V(T)V(T) in ascending powers of TTT, up to the term in T3T^3T3, is 1+16T+CT2+DT3\displaystyle 1 + \frac{1}{6}T + CT^2 + DT^31+61T+CT2+DT3 for ∣pT∣<1|pT| < 1∣pT∣<1.
(i) Find the value of ppp.
(ii) Find the values of the constants C C\,C and DDD.
Use the expansion to find an approximate value for 1.13\sqrt[3]{1.1}31.1, showing your working and giving your answer to 6 decimal places.
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.