<s>[INST] If $T_1=7^7,T_2=7^{7^{7}},T_3=7^{7^{7^{7}}}$ and so on, what will be the tens digit of $T_{1000}$? $7^4$ ends with $2301$, so $7^{4k+r}$ ends with $7^r$ digits
$7^2\equiv1 \mod(4) $, and $7^7\equiv 3 \mod(4) $
Can we use the modulo function in exponent form? I think we will use these two properties, how can I proceed further?
 [/INST] As said in comments observe that $T_n = 7^{T_n}$.
We use induction to find tens digits of $T_n$.
Observe that $T_1 = 43 \mod 100$.
Assume that $T_n \equiv 43\mod 100$. So write $T_n = 43 + 100k$ for some $k$
So $T_{n+1} \mod 100 = 7^{T_n} \mod 100 = 7^{43} 7^{100k} \mod 100$.
Now see that $7^4 = 1 \mod 100$ so we have $7^{100k} = 1 \mod 100$ and $7^{43} \equiv 7^3 \equiv 43 \mod 100$. So we get $T_{n+1} \equiv 43 \mod 100$
So we have $T_n \equiv 43 \mod 100$ for all $n$. This means that tens digit of $T_n$ will be $43$.
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