The initial phase of a biological growth process is modeled by the function B(t)=(2+κt4)7B(t) = \left( 2 + \frac{\kappa t}{4} \right)^7B(t)=(2+4κt)7, where ttt is the time in hours and κ\kappaκ is a non-zero constant.
Find the first 4 terms, in ascending powers of ttt, of the binomial expansion of B(t)B(t)B(t). Give each term in simplest form.
Given that, in the binomial expansion of B(t)B(t)B(t), the coefficients of ttt, t2t^2t2 and t3t^3t3 are the first three terms of an arithmetic progression,
find, using algebra, the possible values of κ\kappaκ.
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.