In a materials science laboratory, the expansion factor of a synthetic polymer PPP relative to temperature change ttt is modeled by the function:
P(t)=(2+kt4)5 P(t) = \left(2 + \frac{kt}{4}\right)^5 P(t)=(2+4kt)5where kkk is a non-zero constant.
Find the first 4 terms, in ascending powers of ttt, of the binomial expansion of P(t)P(t)P(t). Give each term in its simplest form.
To account for sensor drift, a corrected expansion model R(t)R(t)R(t) is defined as:
R(t)=(5+3t)P(t) R(t) = (5 + 3t)P(t) R(t)=(5+3t)P(t)Given that in the expansion of R(t)R(t)R(t), the coefficient of the ttt term is equal to the constant term, determine the value of kkk.
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.
