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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."

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    Dear amWhy.Start from the relation kPn=k!/[k-m]! this formula gives the permutations of k elements m at the time.Keep m=2 aways and k increasing from $3$ to infinity.F$r$om there is up to you to do the rest of the computations.2012-11-26

1 Answers 1

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Let $a_n = \frac{1}{2}(n^2+3n+4)$ and $b_n=\frac{1}{2}(n^2+5n+4)$. The sequence you give is $a_1,b_1,a_2,b_2,\ldots,a_n,b_n,\ldots.$

Thus we either need $\frac{b_n}{a_n}=5$ or $\frac{a_{n+1}}{b_n}=5$. It would appear that there is no positive integer $n$ for which either of these holds.

EDIT: This was written to address the original question about the sequence, not the information in the addendum.