1. ## recurrence problem...

do u have any idea on this?

question :

let n >=2, 0 < X1< X2 <....< Xn-1 < Xn,
for Xk , k=1,2,....,n to be integers

Define
Sn=(1 + 1/X1)(1 + 1/X2)....(1 + 1/Xn) - 1,

then
i) Show the recurrence Sn-1 = Xn/(Xn+1) Sn - 1

ii)If
Sn and Sn-1 are positive integers then Sk is also a positive integer for every k=1,2,3,.....,n

2. i) I think the recurrence is not true.

$\displaystyle S_n+1=\left(1+\frac{1}{x_1}\right)\left(1+\frac{1} {x_2}\right)\ldots\left(1+\frac{1}{x_n}\right)$

$\displaystyle S_{n-1}+1=\left(1+\frac{1}{x_1}\right)\left(1+\frac{1}{ x_2}\right)\ldots\left(1+\frac{1}{x_{n-1}}\right)$

Then $\displaystyle S_n+1=(S_{n-1}+1)\left(1+\frac{1}{x_n}\right)=\frac{x_n+1}{x_n }(S_{n-1}+1)$

$\displaystyle S_{n-1}=\frac{x_n}{x_n+1}(S_n+1)-1\Rightarrow S_{n-1}=\frac{x_nS_n-1}{x_n+1}$

ii) $\displaystyle S_k=\left(1+\frac{1}{x_1}\right)\left(1+\frac{1}{x _2}\right)\ldots\left(1+\frac{1}{x_k}\right)-1$

Every factor of the product is greater then 1, so the entire factor is greater than 1. Therefore $\displaystyle S_k>0, \ \forall k\geq 1$

3. ## tq very much red_dog

u're absolutely right
sorry that i did'nt type the question correctly.

anyway,
thanx a lot!!