The Flower Puzzle Generalized

This article is a crazy generalization of the Flower puzzle by Ankush Goswami published in Gonit Sora on 12th July 2012. We now suppose that instead of three there are $$m$$ temples $$A_{1}, A_{2},dots ,A_{m }$$ and instead of doubling, the flowers increase $$n$$ times in number instantly. Let the priest come to the temple with $$x$$ flowers and keep $$y$$ flowers in each temple. So after keeping $$y$$ flowers in temple $$A_{1}$$ the priest has $$nx-y$$ flowers which instantly become $$n^{2}x-ny$$ ; after keeping in temple $$A_{2}$$ number of flowers become $$n^{3}x-n^{2}y-ny$$ and so after keeping in the $$m$$ th temple number of flowers become $$n^{(m+1)}x-n^{m}y-n^{(m-1)}y-dots -ny$$ which should actually be equal to zero !! Thus

$$n^{(m+1)}x-n^{m}y-n^{(m-1)}y-dots -ny =0$$

$$Rightarrow frac{x}{y}=frac{1+n+n^2+n^3+dots +n^{m-1}}{n^m}$$

$$Rightarrow frac{x}{y}=frac{n^m-1}{n^m(n-1)}$$

The above relation gives all possible values of the number of flowers brought by the priest ($$x$$) and the number of flowers given by him in each temple ($$y$$) .

Debashish Sharma, JRF, Dept of Mathematics, NIT Silchar.