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 AQA A Level Maths 1.7 D: Sequences and series, 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). Each one has a worked solution and a mark scheme showing where the marks go.