Problem link :
Problem
Approach:
Let a_{1} = expected additional waiting time if we have not just tossed the number of six, a_{n} = expected additional waiting time given we have tossed n−1 sixes at t=n−1. we can solve the following system of equations recursively to obtain a formula of the expected time to obtain n consecutive throws of any number in the dice:
a_{1}=1+\frac{5}{7}a_{1}+\frac{2}{7}a_{2}
a_{2}=1+\frac{5}{7}a_{1}+\frac{2}{7}a_{3}
…
a_{n−1}=1+\frac{5}{7}a_{1}+\frac{2}{7}a_{n}
a_{n}=1+\frac{5}{7}a_{1}
Using the properties of geometric series and substituting a_{n}, we obtain the following expression for a_{1}:
a_{1}=\frac{7}{5}(\frac{7^{n}}{2^{n}}-1)
Solution:
Solution to the problem can be found here