Recurrence k-th pattern I am trying to solve this recurrence $T(n) = 6 T(\frac{n}{3}) + n$.
1st recurrence: $6^2T(\frac{n}{3^2}) + \frac{6n}{3} + n$
2nd: $6^3T(\frac{n}{3^3}) + \frac{6^3n}{3^2} + n$
3rd: $6^4T(\frac{n}{3^4}) + \frac{6^6n}{3^3} + n$
I am having trouble describing the general pattern after the k-th iteration.

I assume you are looking to find $T(n)$ after K iterations. First
$$ T(n) = 6 T(n/3) + n$$ then rewriting $T(n/3)$ in terms of $T(n/3^2)$ we conclude:
$$
T(n) = 6^2 T(n/3^2) + 6n/3 +n
$$
similarly $T(n)$ after K iterations becomes:
$$
T(n) = 6^K  T(n/ 3^K) + \sum_{i=1} ^K 6^{i-1}n/3^{i-1} 
$$
or
$$
T(n) = 6^K  T(n/ 3^K) + (2^{K}-1)n
$$
