Question 13
Given a positive integer $k$, let integers $a, b$ (not both zero) satisfy $0 \le a \lt k$, $0 \le b \lt k+1$. Define the sequence $\{T_n\}_{n \ge k}$ as follows:
$$
T_k = a, \quad T_{k+1} = b,
$$
and for $n \ge k + 2$,
$$
T_n \equiv T_{n-1} + T_{n-2} \pmod{n} \quad \text{with} \quad 0 \le T_n \lt n.
$$
Concatenating the decimal expansions of $T_k, T_{k+1}, \dots$ after the decimal point yields an infinite decimal $x = 0.T_kT_{k+1}\dots$. For example, if $k = 66$, $a = 5$, $b = 20$, then $T_{66} = 5$, $T_{67} = 20$, $T_{68} = 25$, $T_{69} = 45$, $T_{70} = 0$, $T_{71} = 45$, $T_{72} = 45$, $T_{73} = 17$, etc., so $x = 0.52025450454517\dots$.
Prove that $x$ is irrational.
$$
T_k = a, \quad T_{k+1} = b,
$$
and for $n \ge k + 2$,
$$
T_n \equiv T_{n-1} + T_{n-2} \pmod{n} \quad \text{with} \quad 0 \le T_n \lt n.
$$
Concatenating the decimal expansions of $T_k, T_{k+1}, \dots$ after the decimal point yields an infinite decimal $x = 0.T_kT_{k+1}\dots$. For example, if $k = 66$, $a = 5$, $b = 20$, then $T_{66} = 5$, $T_{67} = 20$, $T_{68} = 25$, $T_{69} = 45$, $T_{70} = 0$, $T_{71} = 45$, $T_{72} = 45$, $T_{73} = 17$, etc., so $x = 0.52025450454517\dots$.
Prove that $x$ is irrational.