Let
![T_k = k - 1](/media/m/8/7/f/87fd88a2f30f1cb4a422c462aa6f7901.png)
for
![k = 1, 2, 3,4](/media/m/9/7/9/9793e60d5a5e2494c5b9224d29ca2de5.png)
and
![T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).](/media/m/7/f/e/7fe48d388935a31e01d5ecc57b9f527c.png)
Show that for all
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
,
![1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],](/media/m/6/a/9/6a9da69111a0b8b7fb9e7d7f4579e3fe.png)
where
![[x]](/media/m/6/a/4/6a47dfb91475b9d5490dbb3a666604a3.png)
denotes the greatest integer not exceeding
%V0
Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and
$$T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).$$
Show that for all $k$,
$$1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],$$
where $[x]$ denotes the greatest integer not exceeding $x.$