I have been traveling, so responded late. 

Here is a more complete solution for 27:

The following inequalities are true for all real numbers $a,b,c,d$: \[\def\arraystretch{2.2} \begin{array}{rcl} 2a^2 + \dfrac{b^2}{2} \geq 2ab, \\ \dfrac{3}{2}b^2 + \dfrac{3}{2} c^2 \geq 3bc, \\ \dfrac{c^2}{2} + 2d^2 \geq 2cd. \end{array} \] Adding the inequalities, \[ 2a^2 + 2b^2 + 2c^2 + 2d^2 \geq 2ab + 3bc + 2cd, \] and equality occurs when $b=c=2a=2d$. Therefore the minimum value $M=2$.

The coefficients were found by using the parameter $k$ like suggested before.

For 28, what you did is pretty good; to estimate $a_{91}$, the following can be done: