A structural stability index SSS of a micro-beam is modeled as a function of its lateral displacement vvv by the expression S(v)=(2−14v)6S(v) = \left(2 - \frac{1}{4}v\right)^6S(v)=(2−41v)6. Find the first four terms of the binomial expansion of S(v)S(v)S(v) in ascending powers of vvv.
In a symmetric calibration test, the total response is defined as R(v)=(2−14v)6+(2+14v)6R(v) = \left(2 - \frac{1}{4}v\right)^6 + \left(2 + \frac{1}{4}v\right)^6R(v)=(2−41v)6+(2+41v)6. Given that vvv is sufficiently small such that terms in v4v^4v4 and higher powers of vvv may be neglected, show that
R(v)=A+Bv2 R(v) = A + Bv^2 R(v)=A+Bv2where AAA and BBB are integers to be determined.
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.