Question 9
(POL 1) Let $T_k=k-1$ for $k=1,2,3,4$ and
$$T_{2k-1}=T_{2k-2}+2^{k-2},\qquad T_{2k}=T_{2k-5}+2^k\quad(k\ge3).$$
Show that for all $k$,
$$1+T_{2n-1}=\left[\frac{12}7\,2^{n-1}\right]\qquad1+T_{2n}=\left[\frac{17}7\,2^{n-1}\right],$$
where $[x]$ denotes the greatest integer not exceeding $x$.
$$T_{2k-1}=T_{2k-2}+2^{k-2},\qquad T_{2k}=T_{2k-5}+2^k\quad(k\ge3).$$
Show that for all $k$,
$$1+T_{2n-1}=\left[\frac{12}7\,2^{n-1}\right]\qquad1+T_{2n}=\left[\frac{17}7\,2^{n-1}\right],$$
where $[x]$ denotes the greatest integer not exceeding $x$.