**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