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A.P

Posted May 23rd, 2009 by alpha
in

Prove that if p,q,r $ p\ne q $ are terms (not necessarily consecutive)of an A.P , then there exits a rational number k such that $ \frac{r-q}{q-p}=k $

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answer

On May 24th, 2009 Structure says:

Let $ (a_i)_{i\in Z} $ be an arithmetic progression of ratio $ \rho $. Then

$$ a_i=a_0+i\rho$$
$$\frac{a_i-a_j}{a_j-a_k}=\frac{a_0+i\rho-(a_0+j\rho)}{a_0+j\rho-(a_0+k\rho)}=\frac{i-j}{j-k}\in Z$$

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