A structural stability index $S$ of a micro-beam is modeled as a function of its lateral displacement $v$ by the expression $S(v)={\left(2-\frac{1}{4}v\right)}^6$. Find the first four terms of the binomial expansion of $S(v)$ in ascending powers of $v$.

In a symmetric calibration test, the total response is defined as $R(v)={\left(2-\frac{1}{4}v\right)}^6+{\left(2+\frac{1}{4}v\right)}^6$. Given that $v$ is sufficiently small such that terms in $v^4$ and higher powers of $v$ may be neglected, show that

where $A$ and $B$ are integers to be determined.