Let's have the following sequence of natural numbers $4, 5, 7, 9, 11, 14, 16, 20, 22, 27, 29, 35,...a_n,a_n+1$. Does anyone know which two consecutive terms of this sequence have a ratio $(a_{n+1}:a_n)≈(5:1)$?
Informative addendum:
This problem derives from the following poem, written on a marble tablet that was found recently in archeological excavations in the city of Larissa, Greece.
Oh wonderful Goddess of Luck!
Show me how to make a system of lottery tickets where, for every one winner there are five losers.
She answered, "Start by issuing 9 tickets with 5 losers and 4 winners; the tickets of the winners should have the same combination with 6 different numbers. Then, issue 16 tickets with 9 losers and 7 winner; the tickets of the winners should have the same combination with 8 different numbers. Repeat for 14/11, 20/16, 27/22... losers versus winners, with 10, 12, 14 different numbers for the winning combinations, respectively. If you keep doing this continuously, your wish will be fulfilled."