Fibonacci sequence. The Fibonacci sequence yo, y1, Y2₂, ... starts with yo = 0, y₁ = 1, and for t = 2, 3,..., Yt Yt-1+Yt-2, i.e., the sum of the previous two entries. (a) Express this as a time-invariant linear dynamical system with state xt = t = 1,2,.... That is, find A and C, where Xt+1 = Axt and Yt = Cxt. (yt, Yt-1) and output yt, for (b) Use your matrix expression above and MATLAB to compute and plot the Fibonacci sequence y, for t = 0, ..., 20. (c) Using MATLAB, simulate a modified Fibonacci sequence Zo, Z₁, Z2, ..., which starts with the same = 1, but for t = 2, 3,..., values Zo = 0 and Z₁ Zt Zt-1 Zt-2, i.e., the difference of the two previous values. Plot this sequence z, for t = 0, ..., 20.

Find the third term of the sequence  {2n^2/3^n-1}from n=1 to ∞

Find the limit of the sequence

. Let $n \in \mathrm{N}^{*}$ and the equation:$$C_{2 n+1}^{1} x^{n}-C_{2 n+1}^{3} x^{n-1}+C_{2 n+1}^{5} x^{n-2}-\ldots+(-1)^{k} C_{2 n+1}^{2 k+1} x^{n-k}+\ldots+(-1)^{n} C_{2 n+1}^{2 n+1}=0$$Show that:a) the equation admits as roots the numbers::$\mathrm{x}_{\mathrm{k}}=\operatorname{ctg}^{2} \frac{k \pi}{2 n+1}$, for every $k \in\{1.2, \ldots . n\}$  b) $\frac{\pi^{2} n(2 n-1)}{3(2 n+1)^{2}}<\sum_{k=1}^{n} \frac{1}{k^{2}}<\frac{2 \pi^{2} n(n+1)}{3(2 n+1)^{2}}$ c) $\frac{\pi^{4} n(2 n-1)\left(4 n^{2}+10 n-9\right)}{45(2 n+1)^{4}}<\sum_{k=1}^{n} \frac{1}{k^{4}}<\frac{8 \pi^{4} n(n+1)\left(n^{2}+n+3\right)}{45(2 n+1)^{4}}$  d) $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$  e) $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k^{4}}=\frac{\pi^{4}}{4}$  Please check the attached picture for details. I need complete solution with explanation please. the all fifth subpoints are for the same problem and they are conected between them, but if…