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.
Practise AQA A Level Maths 1.7 D: Sequences and series with exam-style questions for A Level Maths. 267 questions covering 1.7.1 Binomial expansion, 1.7.2 Types of sequence (A-level only), 1.7.3 Sigma notation (A-level only), 1.7.4 Arithmetic sequences and series (A-level only), 1.7.5 Geometric sequences and series (A-level only), and 1.7.6 Sequences and series in modelling (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
