# IEMCO8B- Editorial

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

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